We are lucky at our gym, they have a great kids program. The kids 10-14 can take a class to get certified to use some parts of the gym, and roam around outside childcare. They earn a bracelet with the program that they must wear at the gym to say they are certified. They make a big deal in the class about how this is a privilege and that if they don't respect it, their bracelet can be taken back. They also have family classes, yoga, zumba etc... So we have kids around, but it isn't a problem as they have been planned for. Also, everyone who joins knows that kids will be around. DS loves this as he is so proud of his bracelet.
That being said, before this one was around we were at a less family orientated gym, and I respected the rules as those were the rules. Then when the opportunity came I moved to a better place for us.
If so many people are breaking the rules, and as you are a volunteer is there someone you can talk to about maybe doing some family fitness programing at the times that seem to have the most rule breakers? If there is a time and place for it, then maybe people will use that. Another thought, new signs. If the signs have been up along time, people start to just zone them out. Nice bright pink or yellow signs stating the child policy of the gym might cut some of the problem out.