There's several options,
Cobalt Coast on the north side or Sunset House just south of town - both have shore dives off their property. At Sunset House My Bar is a diver's bar and the Cathy Church U/W Photo school/gallery is on-site.
Cobalt Coast is in a quieter neighborhood about 20mins north of town. Divetech there does some interesting dives, they have a rebreather tryout and an escorted scooter dive along the wall to their other location at Lighthouse Point nearby.
Sunset House is in the departing flight path but that's not usually a problem except during the occasional day flight. You can also walk to downtown - to get to anything from Cobalt Coast you'll want a car.
I've never stayed at either resort. I personally feel the better diving is off the north/nw side. Many of the really good dives are in that area. Sunset House being south of town may not go as far past NW Point to dive them, it's a long way for them. The shore dives are equally interesting, off Cobalt Coast you can get to the North Wall - it's a swim though - Sunset House has the famous Mermaid and a small wreck just off their property.
Compass Point(condos)/Ocean Frontiers is an option on the
much quieter East End. As is the Morritt's timeshare/condo complex - there's a Red Sail (Tortuga Divers) facility on site.
We rent condos along Seven Mile Beach since most of the dive operators run shuttles with daily pickup. Most keep their boats in one of several nearby marina's since the diving off SMB is for the most part too far out to access from shore. There's also some hotel options both on/off the beach. A more affordable option is Comfort Suites, Ambassador Divers is on-site there.
In case you're not aware of it, everything on Cayman is expensive. Boat dives average 2/$100 and lodging and food costs are also high. One advantage of Cayman over the Bahamas is you can shore dive there, the reef off most of the Bahama islands is too far out. Most of the sites have a dive operator on site so you can usually arrange for a guide if desired. Some idea can be found here:
Scuba Shore Diving Site Listing for: Cayman Islands, Caribbean