Like any city with a population of over 60,000 Black Rock City attracts some unsavory types. In general you will find far fewer than a similarly sized city in the default world, but there are some.
For a city this large, the crime rate is remarkably low. It is just too much time, effort, expense, and hardship to attend to attract a lot of criminals who are looking for an easy score. But there are some that do show up because our trusting nature makes us appear to be easy pickings. That is deceptive though - burners typically look out for each other and rush to help.
Conventional wisdom is to not bring anything to the burn that will upset you if it is lost, stolen, or destroyed. Lost or destroyed are more common than theft since the playa is really rough on almost everything it comes in contact with, and if you like to get altered losing things is pretty easily done.
Tip - Get to know your neighbors! Neighbors looking out for people they have never seen in a camp before is your first line of defense against theft.
By far the most common crime at the burn. In many cases this is simply a matter of mistaken identity - bikes are easily confused with each other, especially when you are in an altered state. They say that making your bike look unique will help prevent this, but the one surefire way to make sure your bike is there when you return is to lock it every time you get off it. Even a $3.00 cable lock through your wheel and around the frame will prevent someone from casually riding off on it. In camp we use a 15 foot cable lock attached to the vehicle to lock our bikes to.
The other aspect is pro bike thieves - in spite of warnings some folks insist on bringing expensive bikes. The professional bike thieves will be on the lookout for these and have the means to cut your locks. Pros will rarely bother with the inexpensive cruiser bikes that most of us bring though.
Leaving small valuables laying around in plain sight (Ipods, wallets, money, stereo equipment, etc...) may result in their loss. Generally keep these things locked in your vehicle when not in use.
Sometimes inexplicable thefts occur - in 2011 someone stole over 30 bottles of water from our cooler (inside a tent) in the first 4 days. We had so much extra that we could easy have given them gallons if they had asked, but who knows what goes through some people's heads.
Although it is not common in BRC, roofies and other forms of dosing can and do happen. Take all the same precautions you would when in the default world. Use a cup with a cover, don't leave a drink unattended, turn down drinks if the vibe seems sketchy to you, etc...
If you do get dosed, don't hesitate to call for help. The vast majority of burners will rush to the rescue.
There is very little of this, but it can happen - be loud and call for help.
The night of the man burn is what the professional thieves are really there for and most of the events thefts occur during the burn. The city empties out to attend the burn and the thieves pretty much have a free run. Even if you left your shit laying around all week unlocked - lock everything of value up on burn night.