If you’re seeking a quick jog central to downtown, the Nassau waterfront is a good option. The section between Malcolm’s Park and the Harbour Bay Marina is the safest, as you can stay off the road by hugging the paved waterfront walk, but it’s pretty short.
For a longer run, you can also head northeast and cross over the Sir Sidney Poitier Bridge (hilly!) to Paradise Island and follow Paradise Island Road out to Smugglers Beach, for a 7.5 mile route. The Atlantis resort has running paths for its guests.