[I'm not a teacher, mind you]
I think it's safer to say "The curtain fell down". "The curtain fell" makes me think of theatres and such.
We need more context on this, but:
"fall" can be a verb and a noun (meaning autumn). when you say "fall down", everybody knows you're not talking about a season :)
Fall (as a verb) can also have other meanings, including "to fall down". I think a dictionary will help, if you're careful enough.