When did you consider your child(ren) old/mature enough to use public transit? How did you go about making the decision about permitting this type of freedom?
I think this is something that is done gradually.
Thinking back to when I was a kid (my daughter is still too young for public transit), my dad used to take me with him ever weekend into town on the underground. He taught me how to read the maps, and understand changing trains. At some point he started letting me plan the route to our destination and say when we should get off, lead us through stations to the next train etc.
He also started letting me carry my own ticket at some point, and go through the ticket gates by myself.
Eventually, after a year or so, I wanted to go to a friend who lived two stops away, they give me a couple of pounds and sent me on my way.
I think I was about 9 when we started.
Basically, once they can cross streets properly and safely, and don't randomly lose stuff from their pockets, you should be fine. At worst, they can ask an attendant or driver the way.