No, they don't, and the ones that were once prone to had all become Republicans in the realignment beginning in 1964 that was largely complete by the 1990s.
Democrats might argue Constitutional limits on the power of the federal government, but “state’s rights” has been attached to a very specific ideology since before the Civil War.
States' rights do not apply in the opposite case when a fedral rule is made and the states don't want to enforce it or have a law that contradicts it. Their laws can't undermine the fedral law.
Hope that's helpful for your understanding of why this is different to the normal "states' rights" argument which is invoked when a state wants to avoid fedral rules.
They happily do not include science in their political calculus.
I think this should be the key takeaway: it shouldn't be us vs them. Decent people exist on both sides, but they don't seem to run the show on either side :-/