Last year, Democrats flipped one or more chambers in several states, including Michigan and Minnesota, and they have acted quickly, passing measures to tighten gun laws, set limits on carbon emissions, increase education funding and protect abortion access.

The States Project has had a central role. The group, founded six years ago by Adam Pritzker, a businessman and major Democratic donor, and Daniel Squadron, a former New York legislator, has sought to focus its ample resources and attention exclusively on state legislators, trying to fill the void on the left.

“Going back to 1972, the right had seen the extent to which state legislatures were a place that they could impose their worldview,” Mr. Squadron said, noting that Heritage and ALEC were both founded the next year. “The fact that there’s no glamour, and you’re not going to get a presidential candidate sitting on your living room couch by doing this work, the fact that you’re not going to be the top rung of the Beltway, didn’t matter to them structurally, because the return was just too good.”