Check out this article by George Mason University professor Nelson Lund, which addresses that very question. (H/T Robert G.)

Excerpt:

The Supreme Court is hearing two cases this week that represent a challenge to one of the oldest and most fundamental institutions of our civilization. In Hollingsworth v. Perry and United States v. Windsor, the court is being asked to rule that constitutional equal protection requires the government to open marriage to same-sex couples.

The claimed right to same-sex marriage is not in the Constitution or in the court’s precedents, so the court must decide whether to impose a new law making marriage into a new and different institution. The justices are unlikely to take so momentous a step unless they are persuaded that granting this new right to same-sex couples will not harm children or ultimately undermine the health of our society.

A significant number of organizations representing social and behavioral scientists have filed briefs promising the court that there is nothing to worry about. These assurances have no scientific foundation. Same-sex marriage is brand new, and child rearing by same-sex couples remains rare. Even if both phenomena were far more common, large amounts of data collected over decades would be required before any responsible researcher could make meaningful scientific estimates of the long-term effects of redefining marriage.

[…]The prominent National Longitudinal Lesbian Family Study, for instance, relied on a sample recruited entirely at lesbian events, in women’s bookstores and through lesbian newspapers. Other studies relied on samples as small as 18 or 33 or 44 cases. The effect of parenting by male homosexual couples remains in the realm of anecdotes. Most research has relied on reports by parents about their children’s well-being while the children were still under the care of those parents. Even a social scientist should be able to recognize that parents’ evaluations of their own success as parents might be a little skewed.

In 2012, sociologist Loren Marks conducted a detailed re-analysis of 59 studies of parenting by gays and lesbians that were cited by the American Psychological Association in a 2005 publication. Mr. Marks, who teaches at Louisiana State University, concluded that the association drew inferences that were not empirically warranted.

There has been only one study using a large randomized sample, objective measures of well-being, and reports of grown children rather than their parents. This research, by Mark Regnerus, a sociologist at the University of Texas Austin, found that children raised in a household where a parent was involved in a same-sex romantic relationship were at a significant disadvantage with respect to a number of indicators of well being—such as depression, educational attainment and criminal behavior—compared with children of intact biological families.

One might expect this work at least to raise a caution flag, but it has been vociferously attacked on methodological grounds by the same organizations that tout the value of politically congenial research that suffers from more severe methodological shortcomings. This is what one expects from activists, not scientists.

In order for us to be comfortable with adopting same-sex marriage, we need to know that there is evidence showing that there will be no long-term harm to society. Gay marriage advocates are the ones who are demanding a change in the law, so can they prove to us why the change won’t do more harm than good? Have they done a good job of shouldering the burden of proof? I don’t think so. And that means that we should be careful with changing the definition of marriage. We’ve already seen how much damage to children that no-fault divorce, (the last change to the definition of marriage), did to society. We have evidence that same-sex marriage would also harm society. We know that some policies are not good for society, and the side that wants the change bears the burden of proof.

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