Sorry to disappoint but I think you can only access the base during the annual airshow in April when the Navy Flight Demonstration Team and the Blue Angels usually perform.
Having said this, you can still see the Key West Naval Air Station up close. If you visit
Key West you may spy the Combined Bachelors Quarters of the Air Station with the words “Fly Navy” printed on the side, which it is the tallest building on the island and visible from a long distance.