I think that America's done a great job with civil rights, comparatively. That last word is very, very important. And I also think that you need both parties to enforce civil rights. Republicans prevent the government from infringing on our rights and Democrats utilize the government to uphold our rights. Well... ideally. There have been slip ups, but that's the idea, and it's a good one.
Right now, Republicans are doing a lousy job of supporting civil rights. So, right now, I'd say Democrats. But the very act of making civil rights a partisan issue is a major part of why Americans are so angry at each other right now. It's very stupid.