Dubai has one of the lowest crime rates in the world which is an immense source of pride to Emiratis in general and the government in particular. This is largely because of the fact that the UAE enjoys a high standard of living coupled with strict sentencing should a perpetrator be caught committing an act against the law.
However, even with a low crime rate it still makes good sense to stick with sensible anticrime measures that you would normally carry out in your home country, such as leaving valuables and passports in the hotel safe and being discreet when handling cash in crowded areas etc.