My cruise had hundreds of children onboard, but I was aware of them only rarely (I came upon the younger group coloring in one of the stairwells and upon the older group at a pizza party).
The Fun Zone is divided by ages between 3 and 7, and 8 and 12. Off Limits is for teenagers 13 to 17 and features a video arcade, soft furniture and foosball. I stopped by several times, and it was always empty. I did, however, see well-behaved groups of teenagers at many of the adult activities.
Children's activities are scheduled morning, noon and night, but parents are expected to take their children for meals. Princess does not provide private babysitting, and group sitting ends at midnight.