Aircraft Spotting Pages - NAS Key West, Florida
The above map, with modifications, was provided by Google Earth.
About the Spots and What You Will See
At this point, there are no known spots around NAS Key West where you could go spotting, except if you have a boat.
As far as what you will see, NAS Key West plays host to a permanent detachment of F-5 Tiger IIs from NAS Fallon, NV. You could see SH-60s and C-12s occasionally flying to and from the base as well as Hornets from time to time on temporary leave from their home base.
I would highly recommend not trying to find a place to spot around NAS Key West, whether you have a car or a boat. Navy Police, Florida Highway Patrol, and the US Coast Guard would not look too kindly to spotters and people taking pictures of airplanes down there. You're better off going fishing.