It's been a few years since I hit Costa Maya, and the cruise lines were frantically building resorts eveywhere. There was a very nice pool area right at the port, with a very large saltwater pool. Shops and restaurants circling the pool area. There was a small beach, but not very good, very rocky at the time I was there. Young kids will be very content to hang around that pool area all day. Older ones, well , a lot do and there is shopping right there as well.
If you feel adventurous, you can catch a cab into the the town area, but it is very small.
I did and stopped at a small bar area that had it's own beach. You could order a drink, or soda, and they let you have the use of their beach area, which was much nicer than at the port.
Cozumel did have a beach area close to the port area, but it was private/admission. Of course this was four years ago, and a lot of things have probably changed. Most of the people who wanted to lounge on a beach went to the National park, Chankanaab. If you don't want to pay for the excursion there, you can catch a cab to it from the port. There is an admission fee, but I highly recommend it as a place to try. Course if you have already been out on an excursion, you might not have time to go there too.