I have friends who live aboard at Demon's Landing in downtown St. Petersburg. The rates are reasonable (for Florida) but I don't remember the exact fee. They managed to talk the manager into paying the non-liveaboard rate since they're not there every night. For downtown St. Pete, it's an excellent location.
+1 on the beaches! They're about a 15 minute drive from downtown and there are a couple of marinas on the ICW
and Boca Ciega Bay close to the beach. For example, Blind Pass Marina is a 5 minute walk to Sunset Beach.
It's a trade
off. Downtown, you're right on the bay with great sailing just outside the breakwater plus nightlife. The beaches are more quiet (except during spring break) and you might need to pass through 2 draw bridges to get from a marina to the Gulf.
Either way, St. Pete is a great place to spend some sailboat time! Best of luck.