Your pediatrician would have the best answer based on your history.
Personally, I would say you are no more susceptible to stuff than you are here.
My twins first sailed when they were 10.5 months old. Depending on which cruiseline you go with, depends on whether you are eligible for childcare. There are one or two ships down in Florida that have a nursery. "most" others require you to be 3 years old to be in the "kids camp". RCL on some ships offers in room babysitting for a fee, but its hit or miss on its availability.
Anyway, regardless of whether you are in child care or not, cooties are everywhere on the ship...that's why they REALLY want you cleansing your hands all the time. Kiddos in diapers cannot get in the pools unless you are something like Disney with a special system. When you get off at port, you can book your cab to a beach, or book onboard for an excursion. Usually they take you to tame places. In cozumel, we just booked a private cab/van. Another time, we just caught a cab to the beach and had a nice time. Sometimes we just stay on the ship.
My girls are 3.5 and have had 3 cruises with a 4th coming up in December. Its a great way to vacation as a family! I say go for it (after asking your Doc).