Well, I have a lot of experience with this subject, maybe not in America but at least in my own country, and I can say that ... :
your chances of getting a paid job as an uneducated stablehand are pretty slim. On the other hand, this is exactly how I started out, and I kid you not!
When I was nine, I started going around to the various racing stables and asking if they needed help with anything. I was young and crazy about horses; willing to do anything just to be close to them. I got a job mucking out stalls at this old harness racer guy's stable. He was mean and abused his horses, but that's another story. The deal was that if I came to muck out pretty much every day, I'd get to groom some horses, and trail ride one of the horses once a week. It was hard work, but I enjoyed it. I gradually started working more; and I tell you, riding a difficult harness racer who wants nothing more than to buck you off, is one of the best ways to learn to ride! Meaning, of course, trail riding and the basics of handling a horse, not the refinedness of dressage. I had to take lessons for that. : )
So when I was eleven, my dad bought me my first horse to share with my elder sister who was sixteen at the time. When I was fifteen, I went back to that stable with the abused harness racers and bought the horse I'd always groomed and ridden. It was a dream come true; but it required a lot of HARD WORK and will! (I never got supported economically by my parents).
So long story short, you can start out unpaid and gradually work your way up. Your chances of getting a job that way are pretty big. But paid, while untaught? Slimmer....
I'd recommend it though, it's really giving. If you're willing to work.
EDIT:
Mello1: ....bullshit.
I walk behind horses all the time. Right behind their butts. The difference is walking behind a horse when you don't know them or walking behind a horse you know is happy.
The rule goes:
NEVER startle a horse by walking behind it. Now THAT is dangerous. Or walking behind an angry or frightened horse. Always keep a hand on their butt when you walk behind them, so they know you're there.
And -- yes you can walk between a mother and a foal without problem. You can also put a halter on the foal and lead it into the stable. The mother will DEFINITELY follow, but not attack you.