Objection 1: Same-sex marriage violates biblical standards for marriage.
Reply: The problem with Bible verses is that our society is supposed to be a society for everyone regardless of religion, so while the Bible might give religious people a reason not to have same-sex marriages, it doesn't really offer a reason why the secular government might not have other reasons for sanctioning same sex unions for governmental purposes. You have to keep in mind that many heterosexual marriages fail to meet biblical standards, see especially Luke 16: 18. Is the government wrong to allow marriages which were initiated in an adulterous affair? Legally, if you have an affair with the Playboy centerfold, then file for divorce, you can marry the centerfold legally. Biblically, that's not marriage, it's adultery. So, if you want to impose biblical standards on the government, you can't logically stop with homosexuals, you have to de-legitimize all adulterous marriages.
Objection 2: Same-sex relationships are morally unacceptable.
Reply: The state is concerned about what relationships to recognize when, say, end-of-life decisions are made. Should these be made by a lifelong gay partner, or should other family members who were less close to the terminally ill person have the say in what end-of-life decisions this person should have made on his behalf? That the state sanctions a marriage does not imply that the state morally approves that marriage, for the reason cited above.
Objection 3: Same-sex relationships do not produce children.
Reply: Should couples who intend to be childless be allowed to marry? Should women long past menopause be allowed to marry? These marriages will produce no children.
Objection 4: Legalized same-sex marriage will lead to the legalization of every other kind of relationship, including man-boy love relationships.
Reply: Slippery slope arguments are considered a fallacy. Just because we could go further in legitimizing relationships doesn't mean that we should. In the case of relationships with children, the children are thought not to be consenting partners. So we have a principled reason for rejecting, say, a man-boy relationship while accepting same-sex relationships so long as they are consensual.
These are just some things that can be said against the typical arguments against state-sanctioned same-sex marriage. (Church-sanctioned same-sex marriage is another matter). Are there better arguments against SSM?