The Balinese have a reputation as extremely honest people. Unfortunately, not everyone on Bali is Balinese and there has been an increase in petty crimes - especially in Kuta. Nonetheless Bali is still far, far safer than most large cities in this world.
Pickpockets, car break-ins and drive by bag snatching seem to be the most common complaints. To reduce your risk, use plain common sense and take the same basic precautions you would if you were visiting any big city; keep a tight grip on your purse or camera, Never leave any bags (whether valuable or not) unattended in your car, on the beach or by the pool, use the hotel safety deposit box, don’t go down dark deserted gangs (alleys) and don’t count money in the open. Not so hard...
On the flip side don't even think of committing a crime in Indonesia. You are a long way from home with far fewer rights than you think. Behave yourself and be a gracious guest.