You'll need a taxi to take you to a "quiet beach" but there are only a couple of them and they won't have any beach bars or restaurants nearby. You are likely to be the only people visiting these beaches on any given day - they re not among the "better" beaches on the island.
North Friars Bay (not to be confused with South Friars I mentioned in my earlier response) won't have anybody on it but it's on the Atlantic side with very rough waves and strong currents - not really suitable for swimming.
Majors Bay and Turtle Beach are at the southern tip of the island about a (30 minute ride by taxi from the cruise ship dock. They are OK swimming beaches but that's about it. From Turtle Beach (which is very small) you can walk about 3/4 mile over to Cockleshell where you'll find several beach bars and a restaurant.
Whitehouse Bay is 3/4's way down the southern peninsula, 20 minutes or so ride from the cruise ship dock. Salt Plague Bar is located there but it is only open in the evenings. You'll be likely to find a few "sailboaters" using this beach.
There are a few beaches at the north end of the island but they are quite far away (45 to 50 minutes by taxi) from the cruise ship dock and, quite frankly, not the best on the island. THERE ARE NO FACILITIES nearby.
You WILL NOT fiind any transportation at any of the beaches mentioned above so if you plan to visit any of these make sure you bring your own shade (they can be very sunny), plenty of water, food if you want to have something to eat and YOU MUST make prior arrangements for a taxi driver to pick you up at a pre-determined time for your ride back to your ship.
Honesty, for day tripping your best bet is to go to Frigate Bay, South Friars or Cockleshell if you want to spend some time at the beach and have access to food, drink and easy transportation.
