I think that you would want a Disney cruise.
Carnival has a thing called Camp Carnival. Camp Carnival is a year-round, fleet-wide program that provides daily fun-filled and age appropriate activities for children and teenagers between the ages of 2 and 15 years old. In addition, there is supervised "free play" and babysitting service. The play room on each Carnival ship comes completely stocked with toys, games, and puzzles for children for all ages. In addition, each ship has an electronic game room filled with all of the latest video games.
Toddlers (ages 2-5) -- Splash pool, fund family games, late night munchies, cartoon time, little tykes toys, learn your colors, can you count, sponge painting, indoor beach parties, mousercize, pizza pig out, drawing contests, sing-alongs, bingo, face painting, arts and crafts, cookie decorating
If that works for your child and you, it should be just dandy.