Hi Billie Jo,
Best thing would be to get a cab and ask the driver, which are the good beaches. A few years ago we stayed in Grand Cayman at the Hyatt Regency and the beach on 7 Mile beach was just beautiful. I think almost anywhere along the beach would be nice. There may be a few hotels that allow day quests, which is what you would want. You may also want to get out to Sting Ray City and snorkel with the sting rays, its awesome.
Here is a link to an excursion company that does a good job taking cruise passengers out to Sting Ray City:
My family took the route of asking a taxi driver to take us to a beach and ended up at the rundown, dirty, roped off piece of beach so full that there was hardly a place to put your towel down. There were corals and areas for specific ships and groups. It was not a very relaxing experience. Has anyone had experience with paying for the use of one of the more upscale hotel grounds for the day? This may be a solution.