Welcome aboard, Martina! The best beach day of your
Disney Cruise will certainly be at Disney's private island paradise,
Castaway Cay. No need to book a Port Adventure there, as so much is already included on the island. Family Beach is full of beach chairs, loungers, hammocks, and umbrellas so you can find a good spot to set up camp and relax and play at the beach. You'll also want to take your little one over to Spring-A-Leak splash area where you can enjoy some wet and wild (but mild) fun!
In the other ports, you may consider just walking off the ship to explore the port area. Bring your stroller, and do a bit of wandering in town. In Grand Cayman, there is a really beautiful big beach that would be great for you to enjoy for some of the day. It's easy to get to Seven Mile Beach, which is a popular spot, but does offer some quieter areas. Just ask your driver to take you to a quieter section of the beach. Alternatively, a quick cab ride will also take you to Cayman Kai or Rum Point which aren't quite as bustling as Seven Mile Beach. Just know that you'll need to bring your own beach towels and that other things like chairs and umbrellas may be available, but would be for rental on those beaches.
In Cozumel, as you disembark the ship, there are bike drivers with surreys on the back that will offer to drive you the length of the really long pier. That was a treat in itself when our kids were younger, so consider that as a fun mini-activity that won't cost you much at all. Plus, it'll take you to the end of the pier where there is an entire shopping area attached. Do some browsing for souvenirs and grab a bite at one of the local eateries. All without leaving the port area. It's super simple and a great way to spend a bit of time before returning for nap time for your little one.
I hope those ideas help, Martina!
Marcy