You definetly will see more kids on Royal Caribbean over Princess and Celebrity. But depending when you are sailing will determine how many. There will be more this time of year with Spring Breaks, and you will see more during anytime school is not in session like summer, and holidays. You will find fewer right after school starts in the fall and right after the Christmas holidays when they go back to school. The cruise line has a good program to keep them occupied.
If you go at a holiday time there can be more easily more than 500 kids on Voyager class ships, at other times less than 1/2 that. In spite of that, I have never felt overrun by them in any way. There is a lot for the kids to do, and they are not supposed to be in the bars / clubs at night or in the adult pool area. Also, note that on number of occassions with lots of kids cruising, they enforced a midnight cut-off for kids to be out without a parent (so there really isnt a problem with them running wild all night when you try to sleep).
We took a family cruise on Enchantment of the Seas (Royal Caribbean) because we wanted a kid friendly line. We took the kids out of school the first week of December and there were a lot of families on board, including a family with ten kids!! There were always a lot of kids around the pool area, arcade, rock climbing wall, and areas like that. If you are trying to avoid a lot of families, I would suggest Holland America or Norwegian, because I do not think time of year matters on Royal Caribbean.
Just for information, we were on the Carnival Liberty (3,000+ passengers) in February. We were on the elevator one day with an employee from the children's program. We asked her how many children were aboard and she said 250. The children were not a problem at all on this cruise.
However, she did say that during summmer break then can get as many as 700-800 children per week.