I'm not American, so I have no stake in the game. That said, I am in Canada and we have similar problems with what has been called "negative campaigning" since about the late 80s. It's not as bad as there, but now, well, you should see what the Conservatives just voted in as leader....this clown thinks he's Trump Lite on Ice.
They do it (the GOP) because it works. Negatviity travels well and it travels fast and wide. I once took a marketing class in university and learned that people will spread bad news 11 times more than good news.
It's also often short, pithy and easy to remember and repeat. Oh, and it's so much easier than, you know, having actual ideas or solutions.