Teaching kids how to act around horses is a balancing act.  On one hand, they have to remember not to run up to a horse (especially their hind end) or do anything else sudden and startling -- something kids are, of course, rather prone to doing.  On the other hand, they have to learn how to act authoratively with horses, so that the animals know that the kid is the boss.  This second part is especially important, since kids are much smaller and more vulnerable than adults, and tend to be more hesitant around large animals such as horses.  As a result, they need to work harder to establish that position of leadership with a horse.

The important thing to remind kids of is that they aren't going to hurt the horse if they have to kick it a little harder to get it to go, or give it a light smack for misbehaving.  Often kids are too tentative about cueing or disciplining a horse, and then that horse learns that he can walk all over the kid (or kids).  A great example is the girl I nanny for and the pony she rides in her lessons.  The pony has learned that she is fairly passive and uncertain, and as a result has developed all sorts of bad habits, such as taking his feet away from her when she's picking them, and refusing to transition into a trot when she asks.

Of course, at the same time we have to teach kids to be more aware of what they are doing so that they don't inadvertently scare the horse or teach it bad habits.  Whether you are a kid or an adult, being around horses forces you to learn to be more deliberate in your behavior, to set better boundaries, and to be more aware of the things you do without thinking.