No, they don't. Republicans come up with crazy lies about Democrats wanting to give oestrogen to kindergartens and voters somehow lap it up. Democrats win by ignoring these issues during the election in order to let the Republican happily present themselves as the crazy dick-inspectors and then quietly standing up for LGBT rights after the election.

This is because the right-wing base turns out based on vibes, fear, and emotion while the left-wing base tends to look more to actions than words. A Democratic politician can say absolutely nothing about LGBT issues and still be understood as an LGBT ally by their base without giving free sound bites to be played on Newsmax.