Is it illegal to have sex with a first cousin?

I am honestly unsure that a law would be written on the books with this type of activity in mind as illegal per se. There are two ways that I can gauge if in fact the act would be deemed unlawful.

• Marriage Regulations: Regulations vary state by state, but you're not allowed to marry a relative that's too close in blood. Some states allow cousins to marry once they're past fertile years. In other words, if you cannot have kids, you may be able to marry your cousin. Even with genetic counseling, there is a higher risk of medical complications where incest is involved. In reviewing the state laws in appears that Kentucky seems to be on the stricter end of the spectrum. They not only prohibit the marriage between first cousins but also the marriage between first cousins once removed.
• Incest Laws: If it were a distant cousin, the law would generally not concern itself. When there's a relevant increase in genetic disease, the state tends to get involved. Incest laws again vary by state. In Minnesota, sexual intercourse with a person "nearer of kin to the actor than first cousin" would be considered incest and could be punishable by up to 10 years in prison. In North Dakota, marriage, cohabitation, and sex are all considered legal incest when first cousins are involved, and you could spend 5 years in prison and be fined up to $10,000. Laws against cousin-based marriage can be found worldwide in varying degrees of severity.

This being the case, you would need to check the laws in your state. You may find advocate of cousin sexual relations or the "chemistry of cousins." In general terms, sexual relationships between cousins are likely to face legal challenges.