At my barn, your horse has to be over 6 years old (they prefer them to be 7, but it really depends on the individual horse), and that's about it.
We're not supposed to ride unless we have a parent or guardian (someone over 16, pretty much, or someone who could at least call and get help for you if you fall. If the person is under 18, they usually prefer them to have horse experience, I.e. My 16 year old sister could come and supervise because she has worked with horses) or you are riding with at least 3 other people.
I always was under the impression that the person supervising was there so if you fall and get hurt, someone can catch your horse, hold them steady and get you help. Not necesarily preventing a fall, but helping afterwards.