We did a med cruise with Disney last summer, but didn't bring our daughter, who was 1 at the time for a variety of reasons including, you have to pay for the day care on the ship until they are 3, it was our first time in Europe and we wanted to enjoy it without worrying constantly about her, Europe is not stroller friendly. But, we did see many people who had kids that little on the ship. They seemed mostly miserable because it appeared that they were trying to keep their kids with them as much as possible to avoid those extra babysitting fees. At the comedy clubs, in dinner, at the shows, it is just difficult. Keep in mind that Disney DOES have great facilities and if you can reserve the hours, they will keep the child while you are in port. This could just get really expensive.
A sidenote- I disagree with you Hawkeye about not bringing a child to Disney until they are 3 or 4. My daughter is still 2 and has been 3 times already. I do have to give the caveat that we are HUGE Disney fans and knew how to do the parks smartly for our daughter as well as didn't need to ride everything to feel like we were getting our money's worth because we go so much. But, she did really enjoy it and surprises us with how much she remembers from the trip.
Our last trip in April, when she was 2 and a couple months, a castmember gave her a purple foam crown and called her "princess". The other day, she found a tiny piece of the foam from the crown in the car for the first time since and held it up and said "princess!"