High-altitude town and expensive resort, St. Anton am Arlberg is considered a safe place when it comes to crime rates. The Austrian police and security guards are closely monitored, and there are video cameras almost everywhere. Of course, you should be careful in public transportation, do not leave valuables in plain sight. But overall, it's a safe place.
When it comes to personal safety on the slopes, there are numerous rescue services. Mountains are mountains and you can't be careless. The elevators are regularly checked, well-maintained and quite safe. Each hotel, restaurant, club has its own security. The town is small and it's hard to get lost, and the transportation links are tight enough to make you feel safe.