There may be rules that prohibit a minor from sitting at the bar (i.e. the high chairs at the counter where drinks are served), but that is different from club admission.

The official rule is kids age 2+ are each charged full price to enter, under 2 are free, and sometimes kids are subjected to the normal guest policy of no more than 2 guests*. Note that many memberships (including Amex) will allow spouse plus all minor kids… so if for example you have three kids (all under 18), all three kids plus spouse could get in under what would normally be a two guest limit.