Anti-miscegenation advocates suggest that it is either wrong or imprudent for people of different races to marry one another. The question of prudence may be a practical, pragmatic, and even utilitarian one. We cannot necessarily answer such a question in a dogmatic way. But we can answer the question of whether it is wrong.
First, we can cite the following passage, that states that we rightly consider all men to be of one blood. That is to say, there is not "white" blood and "black" blood - but "human" blood. In that sense there is but one race: the human race.
God that made the world and all things therein, seeing that he is Lord of heaven and earth, dwelleth not in temples made with hands; neither is worshipped with men's hands, as though he needed any thing, seeing he giveth to all life, and breath, and all things; and hath made of one blood all nations of men for to dwell on all the face of the earth, and hath determined the times before appointed, and the bounds of their habitation; that they should seek the Lord, if haply they might feel after him, and find him, though he be not far from every one of us: for in him we live, and move, and have our being; as certain also of your own poets have said, For we are also his offspring.
Second, we can emphasize that while there are ethnic divisions amongst people there is one gospel:
Revelation 14:6-7 And I saw another angel fly in the midst of heaven, having the everlasting gospel to preach unto them that dwell on the earth, and to every nation, and kindred, and tongue, and people, saying with a loud voice, "Fear God, and give glory to him; for the hour of his judgment is come: and worship him that made heaven, and earth, and the sea, and the fountains of waters."
While there are divisions amongst men, there is a unity in the gospel. There is no separate gospel along sexual lines (one for men, one for women), not along national lines (one for Americans, another for Russians, and a third for Chinese), and not along language lines. While language may provide a real division between us, in that we cannot worship together in different languages, nevertheless we acknowledge an invisible unity on earth that will find its completion in heaven.
These points should be emphasized, even though anti-miscegenation arguments are not typically directly challenging the question of whether the gospel is for all men. It is important to acknowledge that there are real (even if artificial and more "providential" than "creational") differences and divisions that exist. It is also important for both sides to recognize that these differences and distinctions can and will be transcended.
Whatever you do, though, do not argue directly against anti-miscegenation based on the following verses:
Colossians 3:9-11 Lie not one to another, seeing that ye have put off the old man with his deeds; and have put on the new man, which is renewed in knowledge after the image of him that created him: where there is neither Greek nor Jew, circumcision nor uncircumcision, Barbarian, Scythian, bond nor free: but Christ is all, and in all.
For ye are all the children of God by faith in Christ Jesus. For as many of you as have been baptized into Christ have put on Christ. There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Christ Jesus. And if ye be Christ's, then are ye Abraham's seed, and heirs according to the promise.
Those are great verses for suggesting that we shouldn't have tribal churches, but they aren't about marriage. If the second one were about marriage, we'd have a real problem because of the whole "male nor female" bit.
Remember, as well, that we have Christian liberty, and marriage is (initially) a voluntary association. A man is allowed to use his personal taste in picking his wife. Likewise, as a father evaluating potential sons-in-law, a man can employ his own or his daughter's tastes in deciding what sort of man she should marry. He need not give every suitor the time of day, just as not every suitor must court every single woman.
Because the Bible does not forbid marriage between "races," or "tribes," or "families," we should not say it is immoral. If someone wants to bind our consciences regarding such a thing, they need to come up with a Biblical argument for their position. On the other hand, if they want to make some kind of practical, pragmatic, or utilitarian argument regarding the practice, they should be clear about what they are doing.
One final point: we are called to love our brethren in Christ. Paul is pretty clear that he has a special place in his heart for his brethren according to the flesh, namely the Jews. It's perfectly natural and acceptable for us to have those kinds of positive feelings towards our relatives. On the other hand, we should be on guard against negative feelings towards our non-relatives. Remember how Paul opposed Peter to his face for segregating himself from the Gentiles. Remember also that if we do not love our brethren in Christ (a brotherhood that transcends all divisions: national, ethnic, racial, and even sexual), then we do not truly love God.
So, any anti-miscegenation advocacy that springs from a hatred or antipathy for members of another race, nation, tribe, or the like is clearly immoral because of its source. You may love your own nation, ethnicity, or skin color, but you cannot use that love as an excuse for hate.