Sorry but international law states that children (under 18s) are not allowed in the casino area. Some casinos are stricter than others about kicking the kids out, but they are all supposed to stick to that. Previously I was casino manager onboard the ships so I can tell you from experience that while well behaved kids would be welcome, it has to be one rule for all. You get a gang of running shrieking kids disturbing everyone and you tell them they can't be in the casino they point to the nice quiet kid in the corner and they say "Oh yeah what about them?". So no kids, full stop.
Most ships though do have lots of stuff for kids and teens, even if the shows aren't that interesting for them, there should be plenty of other (supervised) animation, games, teens disco,kids club etc and they'll make friends anyhow, kids do, and they'll want to hang out with their friends by the pool or in the video arcades or whatever. Check the brochure for your cruiseline, there should be something in there about facilities for kids.