No kids, but we still love cruising with Disney. We've seen several instances where a CM has, very politely yet very "this is not open for debate or discussion"-ly, enforced the rules. And a previous poster was right, you're only going to run into issues with "Entitlement Mom" or "Entitlement Dad".
For instance, we were down in the 'sports bar' area during a college football game on a Saturday night when an E-Dad comes in, dragging three toddlers, pulls up a comfy chair, sits down, puts his feet up, and starts cheering (obnoxiously loud) for his team (I won't say who it was, but it rhymed with "Botre Bame". Within probably 30 seconds, one of the servers had come over and quietly informed him that the bar was for adults only. It took him five minutes to be convinced that the rules applied to him, and another three minutes to gather up his kids and leave, complaining the whole time. Luckily, he did most of his complaining during a commercial break, so nobody threw anything at him, but it was getting close there for a while.
The other unfortunate place where you're going to be knee-deep in the precious little darlings is on some excursions. But again, well-raised children aren't a problem. We went on the 7-hour Tulum ruins trip last year (which was just flat amazing). The people behind us on the bus were great. They had about a seven-year-old. The kid read a book, listened to the tour guide, asked his mom a couple of questions, and basically just sort of went with the flow.
The people a few rows ahead of us, however, needed slapped. They brought a hyperactive 3-year-old who had no more interest in a 45-minute boat ride followed by an hour-long bus ride followed by walking around in the hot sun and looking at buildings than he did in doing math homework. That family had been a couple rows in front of us a one of the shows, and the kid had spent the whole show standing up, waving around a banner, hitting people in the head, talking loudly, and generally being a nuisance, and completely unchecked by the parents. Then we got a good look at mom and dad, and we're pretty sure they were dipping into Junior's meds. I don't know if they were stoned, shell-shocked, or brain-damaged, but there was just nobody home. But the rest of us had to deal with this little monster all day, yelling, running up the bus aisle, bumping into people. So frustrating.
~ Mommasita here, I edited out those derogatory terms...~