They're good about the adult shows and nightclubs (once they see that a kid is present) and not so good about the pool deck. Although not sure it's not perfectly OK for children to walk through the area; in fact, on the Dream anyone who arrives via the forward elevator HAS to walk through the adult pool area. I don't have a problem with the kids walking through as long as they don't stay. But I have seen parents with babies in the Cove Cafe and parents dropping in to buy coffee and chat awhile - accompanied by their minor kids.
Palo is very strict. If someone even looks under 18 they ask for proof of age. I assume Remy is the same way although I've never seen anyone underage kicked out of Remy (have seen that at Palo)
Okay, so they WILL enforce the rules if you point it out to the staff?
They should. But it likely won't always happen.
If you are looking for cruises with fewer children...the really long cruises (14-15 nights) had far fewer children. Those they do have are generally preschool age because older ones usually can't be out of school that long. Otherwise, don't cruise in high summer, spring break or over the winter holidays.
They will only allow adults to sign up for the beverage seminars, but depending on where they are held there may be children in the area (especially on the classic ships when they hold the seminars in Diversions during the day).
I went to a cooking demo for adults only on the Dream. Halfway through a guy walks in with a boy about 10 or 11. They sit down and ask for the food sample, and are given one (although only Dad gets the taste of wine). They eat it, get up and leave. Nobody tried to say anything to them about it being an adult program.
On
Castaway Cay last time I was there, a kid about 11 was allowed to remain at the Serenity Cove bar while Dad went out on the beach to look for Mom. It did take him a long time to find her, but as soon as he came back with Mom the family left.