A recent study suggests they do neither.
I'm inclined to agree with this statement from the article.
I seriously doubt that anyone is out there committing crimes simply because the SYG grounds exist, nor do I think that SYG grounds and their fairly narrowly applicable provisions really discourage criminals.
It might make a good plot for a movie or TV show to have someone try to murder someone in such a ways as to be able to claim SYG defense, but realistically I don't see that kind of thing happening in the real world.
As far as it preventing crime--I have a hard time believing that criminals really care if the guy who kills them goes to jail or not, and at the most basic level, the only thing that SYG does is make it simpler to justify self-defense.