Partina cruising

I'll be on a cruise in January that stops at Grand Cayman. All I want to do is go to Seven Mile Beach. I was hoping there would be an excursion though our ship, but so far there is not. If I were to take a taxi, where would I ask to be dropped off? I don't mind dropping my towel in the sand if there is somewhere close that I could get a drink and something to eat. It would be even better if there were chairs to rent but not a deal breaker.