The beach near Atlantis is Cabbage Beach. You can hop in a cab and head over; the public entrance is next to the Riu. It'll be about $4/person each way. When you're done, it's pretty easy to get a cab. Note that there are no public bathrooms on Cabbage Beach; there are some about ten minutes walk away at the Marina Village shops, or you can go to one of the bars or restaurants in the Atlantis Beach Tower and talk your way in or buy something.
The vendors on the public beaches can get quite pushy, and the surf on Cabbage beach can be extremely strong. If you want a more relaxing beach experience getting a beach day pass from one of the private beaches in the area isn't a terrible idea. The cheapest, AFAIK, is the Blue Lagoon, which was $54 for adults, last I checked, including transportation and lunch. The most convenient to the cruise terminal is the British Colonial Hilton, at about $65 for adults ($60 if you book in advance).