The same argument was made against interracial marriage - that the children would be stigmatized, or harmed in some other way. (In fact, one good first step toward removing the stigma is to stop treating the parents differently under the law.)
I am a person of mixed race, and I'm shocked at how many of the exact same arguments against gay marriage and adoption were also used against interracial marriage and interracial adoption. (It's unnatural; it's against God's will; it will harm the children; it will harm traditional marriages; it will destroy the institution of marriage...) I'm curious if the people making the arguments are aware of this and would also oppose interracial marriage if possible, or if they just don't see the parallels.