I got three stories about how I visited backstage:
Several years ago, my friends and I befriended one of the stagedoor guys at St. James. If you know the setup of that theatre, there are two stagedoors: one that is used for stagedooring and one that's connected to the stage. He was at the latter. Anyways, he would let my friends and I go on stage via that door. He took us on tours of the backstage, show us the props and the onstage band area. We probably went back there like four times, usually bringing along another friend.
At another show, a lady that I babysat for had Facebooked an actor and asked if it would be possible for her and her daughter to visit backstage. He said sure. Unfortunately, her daughter got the flu the week of and the actor was leaving the show so it couldn't be rescheduled. So, my friend and I went to the show & had the backstage tour instead.
At another show, I was at stagedoor just because and I recognize the mother of one of the lead actresses. We had met previously at other shows that her daughter did. Anyways, when the actress came out and after she stagedoored, she was bringing her family backstage but told me to come along as well. We didn't go on stage but to the dressing rooms. It was the day before Opening so she showed her outfit that she was going to wear. It was highly unexpected but very sweet for her to include me.