I've been on Disney, Princess, and Norwegian America. Mind you, I don't HAVE kids. But based on what I know from my trips, I'd say Disney hands down.
As someone who cruises without kids, I can tell you, that although the Disney ships typically have 2-3 times the number of kids on board as the others, you see them LESS because of all the amazing activities Disney has for the kids. Seems like there are half as many kids on the Disney ship as others. You sign them up, decide if they have in and out privledges without you, and they give you a pager so they can page you if needed. For when you do want to spend time with them, there are special family areas, like a family pool (situated in between the kids pool and the adults only pool). Also, there's a portion of the ship that goes adults only after 9pm, but other night clubs that kids are still allowed in at that time. My mom and I find some parents complaining they don't see their kids ENOUGH because of how much fun the kids have at their special stuff. Other parents say thats why they'll only do Disney.