I have seen some indications that he was an advocate of the movement, but nothing I can point to as a smoking gun. Any suggestions? Perhaps those indications are in error? I would be interested to know either way.
Theodore Roosevelt was famously supportive of eugenics, and evidence of this is easy to dig up. But as for FDR, I have not seen evidence of this. The few places where I have seen people (non-historians on the Internet) accuse him of this have very flimsy correlations (e.g., some states passed sterilization laws while he was president — he had nothing to do with that legislation whatsoever; or that his internment of the Japanese had something to do with eugenics — it may have been racist, but it was not eugenics).