Very safe..more so than the usa even...really...because criminals in Arab countries are dealt with harshly. especially when they mess with americans or europeans because the countries don't want "international issues" that may effect business/government relationships.
I do recomend adhering somewhat to the local culture..because you will be more comfortable. basicly don't run around half naked or get publicly intoxicated....
UAE does have issues with sex trades..so avoid being alone around men when possible..like taxi's etc...if you do...be aware of your surroundings, and keep your embassy's phone number stored in speed dial on your phone..very effective in threatening anyone who pesters you....but most likely you will only have to worry about getting the local price for things...that will probably be your bigest challenge...have fun