Just got off the Enchanted Princess.
Kids club hours for sea days are basically 9a-12p, 1p-5p, 6p-10p, with paid sitting available 10 pm - 1 am but need to notify staff before 8 pm that day.
On port days, you can drop them off for continuous care while the ship is docked, where they will bring the kids to the buffet area over the lunch hour.
Art/Craft and education activities are scheduled during operating hours while the ship is not in port; otherwise it's basically free play.
There were only 10 kids aged 3-7 and 8 kids 8-12 on our sailing. That was still enough for both the Lodge and Treehouse to be fully staffed with a full slate of activities so I think the number would have to be really low if they made the decision not to open the youth center.
Heads up - save lots of room in your baggage for all the goodies the kids will get over the course of the cruise!