MSC Lirica for Families
The cruise line actively invites families, particularly in the Mediterranean, where about 75 percent of their voyages promote a kids-sail-free special. There is a children's facility, geared from ages 3 - 17, on deck 11. The ship will staff up if enough children sign up, but it must be noted that one challenge is the multi-lingual nature of MSC's audience (and so there are kids speaking all ranges of languages). There's a 24-hour video arcade.
Organized children's activities are less likely on Caribbean sailings because these itineraries are generally longer and more exotic than the usual.