The importance of the tourist trade has made the Bay Islands relatively crime free. The influx of cash has left locals far more affluent than their compatriots on the mainland, and most don't want to do anything to jeopardize that.
The islands are home to vicious swarms of sand-flies and mosquitoes, the latter carry malaria and dengue fever. Anti-malarials are a good idea, chloroquine is the cheapest and most palatable option. The only way to prevent dengue fever is to avoid being bitten. The species of mosquito that spread this disease are only active during daylight hours, so wear a good repellent when you go out.