Added:
The family room has showers, with no curtains or doors, in alcoves, in a space that has glass at both ends and you can see from end to end, from other spaces in the facility?! That means it IS strictly for rinsing before a swim and only rinsing afterward -- adults can't shower in the "family room" unless they want to risk anyone in the entire club seeing them. That forces adults to either forego a shower and do it at home, or bring kids into the 18-and-up space if the adult needs a real shower. Please note -- I do agree that parents should not do that, but from the family room you describe, this facility or club is not providing an adequately private space for families. I would not have taken my kid, who needs to shower for real after swimming due to sensitive skin, into that family room for her to shower! So I wouldn't have been able to use your club, with my child, at all. It seems you think the family room is fine, but what you describe gives parents no option at all for a shower for children or themselves, since the room is basically visible to everyone in the facility. I'm surprised a family has not called them out on this. Again, if the club wants to be for over-18s only, that's just fine, but they can't expect to tell people kids are allowed and then assume that kids don't need showers, or don't mind taking them in view of others. You want your privacy. Even young kids deserve theirs too, and their parents should be able to have actual showers at the facility without violating the age rules for the adult locker room--From what you describe that does not sound possible.
Original reply:
If this is a membership club -- by which I mean, you pay to belong -- then you can talk to management about whether violation of the rules can result in their telling the violators that their memberships can be ended if they flout the rules repeatedly or if they refuse to do what staffers say. You write that a staffer said she'd told a mom (just "a" mom or this particular person?) not to take a child into the adult dressing room and the mom blew it off and pushed into the room anyway. I would suggest that the staff keep a log of those kinds of things and anyone who does it more than once (or however many times the management chooses as its limit) would be told that further infractions will cause the membership to be revoked. If you see this person with a child in the adult area again, I'd talk with the manager of the whole facility about their policies and how they intend to enforce them. Tell the manager that you are considering moving your membership elsewhere over this and you expect them to enforce the rules they have posted!
I would note that if this is a very busy place, and kids are able to use certain facilities like a pool -- the "family changing rooms" can be very busy. If these rooms only accommodate one family at a time, others have to wait (and possibly wait and wait) for that space to change. It happens at the county recreation centers here; kids are allowed in the changing rooms if accompanied (same-sex only if the kids are above a certain age) but there are also family rooms, yet getting one of those family changing rooms is pretty much a no-go at busy periods. There is a shower and toilet and it's lockable but it's essentially one very large bathroom, not a full multi-familiy changing area. I see lines of families with wiggly little kids getting frustrated standing around in the cold hallway waiting for those designated rooms. So while I'm definitely not defending those who flout the rules at your facility (where it's clearly 18 and up in the adult spaces), I wonder if the family changing rooms at your facility are so backed up that some parents have just given up on them. It would be worth checking out those spaces and telling management, if it seems the issue might be that the family rooms are inadequate. That may not be this one woman's reason for bringing in her child, but it IS a reason why some folks may do so at times, if the family rooms are like the ones I've seen.
As for your feelings about having a young child in the space where you're changing, that's your prerogative, but I would not feel that way at all. If kids are kept from even seeing same-sex adults change, they pick up on the idea that there's something wrong or somehow icky about shared shower spaces and public changing rooms--that will carry over to gym class later in school. That isn't useful. It's not like they're watching the opposite sex roaming a locker room in nothing but towels. I'm not about flaunting things and prefer to be covered even around other adult women, but the presence of a little girl wouldn't bother me; I doubt she'd be really staring at me and if she were, well, a good stare back and a "Can I help you?" will shut down most kids.
And it's not up to me to worry about her seeing other people's piercings or tattoos --that's her folks' job to handle, not mine.