The Dominican Republic is not as violent as many other nations in Latin America, but it's still best to keep a low profile and keep your valuables out of sight, as the nation is not a wealthy one. That should not deter you from exploring, however - just practice common sense.
Tourists are often considered attractive targets for criminal activity. So you'll want to keep a low profile in order not to become a victim of violence or crime.
DO
* make sure your clothing blends in
* walk confidently and don't look at your map in the open
* don't carry a large amount of cash
* stay on the main roads/streets
* carry a copy of your passport and keep a number for your hotel handy
DON'T
* Look like your rich-leave the costly clothes and accessories at home!
* carry around ATM or credit cards if you don't have to
* carry any valuable documents if you don't have to
Never fight a robber; stay calm, cooperate, and then inform the police. The police can be a bit unreliable and may seek bribes, so be careful.