I think the wording of both is fine. I prefer the first. But it isn’t clear what you mean. Most people don’t fall when they run so it seems like a strange thing to say. I would say “Be careful not to fall” in a situation where the consequences or the risk of falling are high - for example hiking on a trail that is on the edge of a cliff.

I think it's more logical to say, "Be careful or you'll fall over (or off or down as appropriate)!" The problem is an imperative negative is more complicated than necessary and means the kid is bound to fall! The second doesn't make literal sense to me, I know what it means but it's totally illogical .. what are you saying? Be careful or you won't fall down? Because it COULD mean that. So whilst it's perfectly possible someone might say it, it's not a good example to copy in my opinion. Native speakers DO say some daft things though .. so you might hear it :)

Either one of those works. I think both sound good for most every day situations.