I did dive trips to Key West and Puerto Rico solo. The Fajardo Inn has a package with Sea Ventures Dive Center (all of which you can find online) that I liked. Fajardo is a little touristy. The P. Ricans like to travel there for the beaches and the hiking around El Yunque. The dive charter is pretty good. The only draw back is that you'll have to rent a car to get to the eastern part of the island from the International Airport in San Juan, and to drive from the hotel to the docks. The surrounding area is prety cool too for post-dive activities: hiking, swimming, kayaking (highly advise the bioluminescent lagoon trips), and just hanging out.
I also advise South Point Divers for Key West. The nice part about Key West is that you can walk to the docks from just about any part of the town where you might be staying. I can't say anything about the price of hotels (I stay at the Navy Lodge when I travel there).
As far as the safety thing, just do your homework before you travel anywhere solo. So long as you have a good map, don't do anything dumb (such as flash your room key to everyone in the local bar), and give yourself plenty of time to make it to the docks for your dive, you'll be cool.
I travel solo alot, and I have never had an issue (even when I lived in Europe). It's all about your mind set when you go.
My moto: travelling solo rocks! You can eat when you're hungry, nap when you're tired, see what you want to see when you want to see it, and meet whomever you want by simply asking them to take your picture.