I'm the family cruise editor here. Both Carnival and Royal Caribbean have excellent youth programs. Carnival's youth programming includes some fun family activities like Family Craft Night and Name that Tune game/family disco party. Royal Caribbean's youth programming is non-stop activities which is great if your kids are high energy. Royal Caribbean also has some interesting programming including Art by Crayola craft projects and Adventure Science, which is fun science experiments. Both lines have very competant youth staff -- parents have to sign kids in and out of the program. So it's really your call as to which to choose. (Both ships should have video arcades for the kids to play for a fee.)
Feel free to ask me more specific questions.
Smooth sailing,
Luisa Frey Gaynor