Safety and Security on Bali
In light of the bombings of 2002 and 2005, nobody should plan a visit to Bali without considering their safety. That said, you can visit Bali, enjoy yourself, and not worry too much about your security and safety. At places that attract large numbers of people, such as shopping malls and even some restaurants, security guards now check all who enter. The security measures are largely unintrusive. At the most, you'll be asked to open your bag, if you're carrying one.
Of course, the best advice is to avoid places that attract a lot of tourists entirely. That's not entirely impossible, since there are many parts of Bali where you can find a good, but not over-built, tourist infrastructure. There's Lovina on the north coast as well as Candi Dasa and even Sanur on the east coast.
If you do want to be in the thick of things and stay in the Kuta / Legian area, then you're advised to be on your guard about potential dangers. Avoid places that are regularly crowded. If you do find yourself in a crowd, be aware of where the exits are. It should go without saying that you should get out of any situation where you don't have more than one exit available to you.
Protecting Your Valuables
There is very little violent crime on Bali, but no place is absolutely safe. Any place that attracts a lot of (comparatively) 'rich' people will also attract those seeking to take advantage of them. Most hotels provide in-room safes. Use them to keep your extra cash and other valuables safe when you don't need them with you.