Find Your Cruise
It very much depends on the ship you are on. If you book a 'child friendly' ship you will be fine. A good choice would be Island Star / Escape. Princess cruises offer some good ships with excellent child facilities including baby sitting services.
One good thing about cruising is that you can get out of the heat if it's getting too much which is essential for babies. If you go out for the day sightseeing make sure you're prepared for looking after baby in the heat - remember young kids don't have inbuilt thermostats to cool them down like we do. If you're happy to leave them behind with baby-sitters, there are plenty of ships that offer this service - Royal Caribbean is one I'm sure.