If the girl is 10, she is old enough to follow rules. But you can't expect non horsey people to automatically know what is or what isnt safe around horses.
Talk to her nicely before you go out to the barn. Let her know that your horses are sensitive, and while they dislike loud and fast movements, they appreciate kind, soft words and actions.
Let her know that she may not run or be loud around the horses (ie screaming or squealing with laughter.) Tell her that you will take her to see them, but she may only ride (if you are ok with that) if she remains calm and listens to you, so as not to upset the horses.
Then lead by example, by not yelling at her, but by remaining pleasant and helpful. If at any time things get out of hand, just let her know that the horses are getting nervous, and a nervous horse can be dangerous and unpredictable.
Try to be sympathetic to the fact that the girl is probably very excited. Remember what it was like to be a little kid, and dying just to be around horses?
Try and get her dad to stay at the house!! Lol