I believe the "enemies of the cross" in Philippians 3:18 are the same type of people who despise the "offense of the cross" in Galatians 5:11. They are not enemies of Christ (at least not in word), but they are enemies of His cross. They are still seeking their own righteousness through good works and/or rituals, instead of submitting to God's righteousness through faith in Christ. In the Apostle Paul's day, these types of people were Judaizers (i.e., legalists) who taught that in order to become a "real" Christian, a person must comply with the Mosaic law. In our day, the spirit of these Judaizers is most commonly found among people who require any type of works prior or subsequent to becoming a Christian. For them, simply believing in Jesus Christ for eternal life is not enough. Rather, they teach that in order to be a "real" Christian, good works and/or rituals must accompany belief. I will now explain why I think Paul was referring to Judaizers in Philippians 3:18.
Let's first look at the relevant passage in Philippians 3:17-21: "17 Brethren, be followers together of me, and mark them which walk so as ye have us for an ensample. 18 (For many walk, of whom I have told you often, and now tell you even weeping, that they are the enemies of the cross of Christ: 19 Whose end is destruction, whose God is their belly, and whose glory is in their shame, who mind earthly things.) 20 For our conversation is in heaven; from whence also we look for the Saviour, the Lord Jesus Christ: 21 Who shall change our vile body, that it may be fashioned like unto his glorious body, according to the working whereby he is able even to subdue all things unto himself."
The first thing we should notice is that in verse 17 the Apostle Paul tells the Philippians to follow him. Why would he have to say this? Who else could the Philippians potentially follow? Paul answers this question in verse 18. Notice that he uses the conjunction "for" to connect the two sentences. Paul is saying, "Follow me as opposed to the enemies of the cross." Paul goes on to say that he has already told the Philippians of them "often." This word "often" reminds us of what Paul said in verse 1: "To write the same things to you, to me indeed is not grievous, but for you it is safe." After saying this, he immediately warns the Philippians of the Judaizers in verse 2 and tells the Philippians that they themselves are the true "spiritual circumcision" in verse 3: "2 Beware of dogs, beware of evil workers, beware of the concision. 3 For we are the circumcision, which worship God in the spirit, and rejoice in Christ Jesus, and have no confidence in the flesh." After this, Paul continues in verses 5 to 9 to contrast his former life as a person who had the qualifications of a Judaizer with his current life as a person whose hope is solely in Jesus Christ. Philippians 3 is really Paul's polemic against the Judaizers. His intent is to make sure the Philippians follow him and not the Judaizers. The last thing he wants is for the Philippians to be deceived in the same manner that the Galatians were. Therefore, it is natural that he ends the chapter with a final warning against them. Let's now examine the relevant verses a bit more.
The statement "whose God is their belly" taken alone makes me think of 2 Corinthians 2:17 where the Apostle Paul tells us about the many people who "peddle the word of God" (NKJV, NASB). For them, preaching may just be a means of feeding themselves. In Philippians 1:14-18, Paul speak of those who preach Christ out of envy, contention, and pretense. It could be that their "God is their belly." But at the same time, Paul was still thankful that they were preaching Christ. In contrast, Paul warns the Philippians against the "enemies of the cross of Christ." Therefore, it is unlikely that those in Philippians 1:14-18 and those in Philippians 3:18 refer to the same group of people. If the expression their "God is their belly" refers to people making money off of the gospel, it still doesn't tell us anything about the content of their message. And that is the biggest issue. Regardless of their motives, what were these "enemies of the cross of Christ" teaching?
The belly could be referring to the "flesh" and this again brings us back to verse 3 where Paul says that unlike the Judaizers, we "have no confidence in the flesh." Paul also refers to "dung" in verse 8, which is a product of the belly. In Romans 16:17-18, Paul talks about those who serve their own belly: "17 Now I beseech you, brethren, mark them which cause divisions and offences contrary to the doctrine which ye have learned; and avoid them. 18 For they that are such serve not our Lord Jesus Christ, but their own belly; and by good words and fair speeches deceive the hearts of the simple." In these verses, it is clear that Paul is condemning doctrine that sounds reasonable and good. It is doctrine that is capable of deceiving the hearts of the simple. The Galatians were deceived by Judaizers in this way (Galatians 3:1), and it may very well be that Paul was warning the Romans of these very people. There are some commonalities between the two letters, especially in regard to the doctrine of justification, so Paul might have been trying to preempt the same problem from occurring in Rome. In Romans 16:19, Paul says that the Romans had been "obedient," and in light of the immediate context, I interpret this to mean "obedient to doctrine." This fits well with one of the letter's purposes, namely, "obedience to/of faith" (Romans 1:5, Romans 16:26). I interpret this to mean "obedience to [the doctrine of justification by] faith." However, similar to Philippians 3:18, it is impossible to say with complete certainty who Paul had in mind in Romans 16:17-18. In light of the context, though, I think it was the Judaizers. Likewise, in 2 Corinthians 11, we learn that false apostles transform themselves into the apostles of Christ, but are really deceitful workers who follow Satan as false angels of light (verses 13-15). This deceitfulness also reminds me of Romans 16:18. These false apostles who preach "another Jesus" and "another gospel" (verse 4) are later identified as Judaizers (verse 22).
It is sometimes said that these "enemies of the cross" were really antinomian-like believers who refused to live the "cruciform" life of denying oneself and following Christ daily (Luke 9:23). Since their "end is destruction" (verse 18), some people claim that these people are doomed to eternal destruction (2 Thessalonians 1:9). However, people who say this fail to distinguish between being a "believer" and being a "disciple." I have written about this in my article What is the Good News of the Bible? Also, Paul himself in Philippians 3:10-14 says that living the "cruciform" life is not a matter of "getting saved," but a matter of having a more excellent resurrection (see the special Greek word used in verse 11 for resurrection) and winning the prize (verse 14). All believers are resurrected to life (John 11:25-26, etc.) and will have praise from God (1 Corinthians 4:5), but not all of them receive the prize (1 Corinthians 9:24-27). In Philippians 3, Paul refers to the "cruciform" life in regard to winning a prize, whereas the "enemies of the cross" are in danger of being destroyed (verse 18). The penalty for not winning the prize is certainly not eternal destruction. In fact, Paul basically admits that there were some people who didn't live this life (Philippians 3:15), and his solution was simply to admonish them by saying "Nevertheless, whereto we have already attained, let us walk by the same rule, let us mind the same thing" (verse 16). Paul obviously wants everyone to follow him as an example and role model (verse 17), but he realizes that not everyone is at his level. Now, I do grant that some believers can be destroyed (i.e., suffer temporal affliction and premature death) in this life due to rebellion and antinomian-like living (1 Corinthians 5:5, 1 Corinthians 11:29-32), but that is only temporal destruction that does not impact their eternal destiny in heaven (again see 1 Corinthians 11:29-32). So, if antinomian-like believers are in view in Philippians 3:18, they may very well be destined for destruction (i.e., temporal affliction and loss of life), especially as the day of the Lord was approaching (Hebrews 11:25). However, in Philippians 3:18, Paul is probably referring to eschatological destruction (i.e., eternal destruction or destruction in eternity) because in addition to saying their "end" is destruction, he immediately goes on to contrast such people with those who are transformed into glorious bodies at the appearing of Christ (verses 20-21). In light of this (granted that my interpretation is correct), it is not possible for the "enemies of the cross" to be antinomian-like believers who fail to live the "cruciform" life. (As a side note, please see my article Explanation and Refutation of "Justification by Experientially Participating in Christ's Death and Resurrection" for a refutation of "salvation by cruciform living." There are an increasing number of people who are teaching this type of "works-based salvation" and some of them are using very subtle language and redefining terms to disguise it.)
People who are saved by grace have nothing in themselves to boast about. The Apostle Paul told the Corinthians that God chose the "foolish things" of the world so "that no flesh should glory in his presence" and that "He that glorieth, let him glory in the Lord" (1 Corinthians 1:29, 31). In contrast, the Judaizers boast in their works (2 Corinthians 11:12, Galatians 6:13). Paul speaks of this boasting in Philippians 3:5-6. If the "enemies of the cross of Christ" were antinomian-like believers, it is hard to imagine that they would have anything to boast about in the context of religious practices or moral living. Perhaps it is imaginable that an antinomian would boast in his freedom to sin, but in the context of Philippians 3, it seems highly unlikely that Paul would suddenly change topics so radically without even giving his readers a hint. Up until verse 18, he had been in the midst of a polemic against Judaizers, so it would be odd, to say the least, if he suddenly started a new polemic against antinomians without even identifying them explicitly.
This statement again makes me think of "flesh." Judaizers boasted in the flesh, whereas Paul and the Philippians had "no confidence in the flesh" and were spiritually minded (verse 3). More than this, however, verse 19 immediately leads into verses 20 and 21 where Paul is talking about the Philippians heavenly expectation. This contrast between earth and heaven reminds us of Paul's allegory in Galatians 4:21-31. In particular, he chides those Galatians who desired to be under the law and refers to such people as being born under bondage after the flesh. In verses 24 to 26, he says the following: "24 Which things are an allegory: for these are the two covenants; the one from the mount Sinai, which gendereth to bondage, which is Agar. 25 For this Agar is mount Sinai in Arabia, and answereth to Jerusalem which now is, and is in bondage with her children. 26 But Jerusalem which is above is free, which is the mother of us all." So, we have this same contrast in Philippians 3 between the earthly and the heavenly. In my opinion, this is strongly suggestive that Paul was referring to Judaizers. Of course, antinomians also "mind earthly things," but if Paul were talking about antinomians here, we have the exact same problem I mentioned previously, namely, the improbability that Paul would have changed topics so suddenly and drastically without giving his readers any hint about it.
Paul's brief description of the "enemies of the cross of Christ" in Philippians 3:18-19 is ambiguous enough to apply to both Judaizers and antinomians. However, when viewed in context and in relation to Paul's other writings, especially Romans and Galatians, I believe that the weight of evidence strongly suggests that he was talking about Judaizers.