Take somebody who's smart, professes generally Good Liberal values, believes themselves to be nontheistic (or at least wholly secular) in how they approach social issues, and wishes to be rational in their political beliefs. Tell them, genuinely, that you believe that, so long as they're consenting adults, blood relatives should be allowed to legally marry. Watch them freak out.
See people who otherwise profess a deep committment to reproductive freedom tell you that blood relatives shouldn't be allowed to marry because of the genetic consequences should they breed. See people who would otherwise oppose the idea that the purpose of marriage is procreation suddenly embrace the idea that procreative consequences should determine who has the right to marry. Do not expect them to be swayed when you point out that marriage is not a license to procreate and that unmarried people have babies all the time.
Let me be clear: there are, to be sure, all sorts people I know who are romantically involved who I would prefer not breed. In some of these cases, I could muster arguments for why their breeding would be, at least from a probabilistic standpoint, genetically undesirable, and these arguments would have a good deal more scientific validity than any abstract claim about what is genetically bad about procreation among blood relatives. And yet, crazy me, I think it would be wrong for me to have laws prohibiting other people who want to get married from doing so just because I don't want them to have children or because I think science says they shouldn't have children.
Check out Drek's post from yesterday about cousin marriage. He gets all the way to acknowledging that letting first cousins marry wouldn't be a gene pool disaster, but yet he's still against cousin marriage. Why? Because he can come up with some narrative about how, over some protracted period of time, allowing cousins to marry would cause the unraveling of Society.
One might note that cousins have long had the right to marry in all sorts of societies that are older than American society and not, as yet, unraveled. Or one could note the ample evidence that, generally speaking, people don't want to date/marry/have-babies-with their cousins, so it's not like it would result in some massive society-unraveling upturn in such marriages. But, more importantly: just because we can come up with some speculative narrative about how preventing two people from getting married might be in the very-long-term interests of society, are we justified in denying people the right to do so? I would have to be pretty confident in the validity of the narrative and the severity of the disaster before I would be comfortable with that.
A newer and deliciously twisted argument against believing that relatives should be allowed to marry is that, in debates about same-sex marriage, conservatives sometimes make the argument that if you give gays the right to marry, people are going to want to allow every conceivable configuration of people, including pairs of blood relatives, the right to marry. So if you say that relatives should be allowed to marry, you would be allowing conservatives to gain rhetorical points by saying "I told you so!", and thereby you would be contributing to the harming the cause of gay rights in America. Or something like that. As if the prevention of cousin marriage is a genuine part of the reason that anybody who opposes same-sex marriage does so. Anyway, don't get me wrong: I am completely fine with a change in laws that just allow same-sex marriage without extending marriage to any-pair-of-consenting-adults, but that doesn't mean that prohibiting people who happen to be genetically related from marrying isn't still ultimately wrong.
Besides which, various arguments just seem like rationalizations for the real reason otherwise-good-liberals don't want to allow blood relatives to marry: they think it is gross. I understand it's gross. I feel I need to issue a disclaimer that I have never wanted to marry and/or procreate with any relatives of mine. Still, I just don't really think "Ew. Ick." is a valid argument for imposing restrictions on the rights other people have regarding the ways in which they can share their lives.