In your example, just "fall" is fine.
"Fall down" is somewhat idiomatic. To a non-native speaker I can see why it is strange, because surely the nature of gravity means everything falls down meaning "fall down" can seem like a tautology. But sometimes you need it.
She fell down the stairs.
This would be used to describe a situation where someone tumbled down the stairs and landed at the bottom. You need "down" because the sentence would be grammatically incorrect without it.
You could say...
She fell on the stairs.
... but this carries a different inference. Maybe they just fell on one step. It doesn't carry the idea that they continued downwards and landed at the bottom.
Even without that much detail though, you will hear statements like:
She fell down.
You'd normally only say "fall down" if you want to include the inference of someone landing on the floor or ground, because of course something can fall, but be caught before it hits the ground. There is no specific need to include the word "down", certainly not in your example, but in some situations it does differentiate between someone just tripping or stumbling but managing to stop themselves or being caught.