I don't know of a minimum age limit. My two kids, now 1.5 and 4 years old, are members of AA and UA's programs, and last year they almost achieved Premier status on UA.
Since the younger one travels on half-price tickets, he sometimes doesn't get miles for his flights. UA explains that the half-price infant tickets are not eligible for miles. However, about half the time, he gets the miles anyway.
UA Mileage Plus CSAs explain that those miles may be subtracted if detected by an audit....