Athletics in general can help curb violence. Especially the more "violent" athletics. That's why so many Juvenile programs use sports like boxing, wrestling, football, etc as part of their rehabilitation program.
It stands to reason that the general public would trend the same way. That is to say, that the more people participate in Combat Athletics, the less they will get involved with violent crime.
The obvious exceptions are the ninjer schools that don't do any kind of sparring or practice anything remotely "athletic" or physical because they teach stuff that is "too deadly" and all of the people that study there are gun toting RBSD maniacs looking for bums to shoot. But luckily, the intellectual bankruptcy of these individuals prevents them from being any real threat to anyone but themselves...
Additionally, criminals are criminals. They will find some way to do what they are going to do. Whether it is by gun, knife, or a "give me your wallet or I'll spinning elbow the **** out of your face!!!" (I had to laugh at the OP's reference to a Muay Thai mugger), they will find a way to commit a crime.
Guns don't take months or years of training to be used effectively. They're used by criminals for that exact reason. If most criminals really had enough dedication to train at anything long enough to be good at it they probably wouldn't need to think about stealing or robbery.