Dinner Key marina has reasonable mooring rates. Depending on your draft, you may fit in one of the small free anchorages outside the main mooring field. I was Dinner Key's first mooring customer ... and put my boat on the mooring that was closest to my old home-built mooring in the formerly free main anchorage. There are good people there, and some not so good. Mooring customers get access to a daytime hourly launch shuttle, a dinghy dock, the laundry and bathroom facilities, and a parking sticker.
Coconut Grove Sailing Club has a mooring field as well, which is actually open to the public. Their launch is on-demand, so more convenient than Dinner Key's.
If you just want to anchor in the Keys, consider Sunset Cove in Key Largo. There are a few boats moored there full-time. It's on the inside so there's protection from the east.
If you do stop in at Dinner Key or CGSC, make sure to make time for lunch at Scotty's Landing. You can get there by dinghy. The cracked conch salad and dolphin sandwich are great and the view can't be beat.