I lived four years in the Caribbean - including 2 in Jamaica - and the precautions are the same you'd take anywhere else. Don't wear flashy jewelry or clothes, don't go into dicey areas, secure valuables like your wallet, passport, camera. Avoid being a livemouth jerk. Be careful of taking free or paid-for rides - only go with taxis and buses recommended by your hotel.

The same sort of precautions I now take while living outside of NY city, or when travelling in the U.S. or Europe. Tourists are prime targets for criminals everywhere, not just in the Caribbean.