Probably a majority of opponents base their opposition on religious grounds, as contrary to church teachings and natural law. There is also a secular argument that deserves more attention than it has gotten.
Marriage, so this argument goes, serves three important social functions: procreation and child rearing, mutual care and assistance, and reining in horny young males by linking approved sex to a serious and long-lasting social commitment.
True, the sexual liberalism that began in the Sixties has undermined this third argument; nevertheless, it's important for the laws and customs to give maximum support to the traditional institution of marriage. At least that support will slow down the slide into totally commitment-free sexual license, resulting in ever more single mothers and fatherless children.
The gays and lesbians may well say, we're pro-marriage, and those are valid arguments. But how does allowing us, loving and committed couples, to enjoy the status and benefits of marriage undermine the institution of marriage?
The response is that prohibiting discrimination against same-sex couples seeking marriage cannot be limited to just those couples. If two men or two women can marry, why not two of each as a foursome? Or a Muslim taking four wives? Or brothers and sisters? Or an entire Perfectionist "complex marriage" commune (Putney, 1830s)?
Gay marriage advocates scoff at this as a red herring. They should however, heed the argument made to the Supreme Court by Vermont Attorney General William Sorrell - certainly nobody's conservative - in the Baker appeal. Said he, "[If the Court accepts gay and lesbian Appellant's argument for] separating the [marriage] statutes from their language and their historical foundations, the groundwork will be laid for other groups to claim the right to marry. The most obvious are polygamists and proponents of group marriage. Following the arguments of Appellants, such persons would have strong claims to fit within the 'purposes' of the marriage statutes."