Since the airport is located near the port, there really is nothing close by. Also, parts of the neighborhoods surrounding the airport are not the greatest. There are no real sidewalks either.
I guess you could grab a hotel shuttle. The car rental ones put you in the middle of nowhere. So they wouldn't help/
If you really wanted to go somewhere, I would walk to US1, head south to find the most stuff. But it is a long walk and I wouldn't recommend it.