Many associate martial artists with being violent. This is so untrue. True martial arts is a practice of self discipline and control. As always, a true warrior can make a friend of an enemy.
So why would a peaceful Buddhist want to practice martial arts? Well, there are a many factors. Martial arts is a great way of teaching oneself self discipline, and training the mind. When training the body, we also train the mind. Constantly focusing and being mindful of the body has many benefits not only in a martial aspect, but also of health. The ultimate goal is achieving 'No-Mind." Zen or mushin.
Martial arts, at an advanced level, incorporates the involvement of Qi or Ki. The energy believed to be within us all, that also binds us all together. Martial arts is an excellent way of focusing Qi (Chi), into various parts of the body for healing, balance, and strength. Being mindful of our bodies and our Qi gives us better communication with that "inner self," as well as others.
As far as action movies loaded with lots of violence, even involving martial arts, I'm not a fan. Though I do collect rare Martial Arts movies from China and Japan, I tend to stick with the ones that have a good moral message or a message about training.
It is my hope that folks don't associate martial artists as violent people, (the ones who truly practic The Way), but rather as peaceful people who know when to act upon a situation when called for. Whether it be something as small as making a decision, or as important as saving your life.