I have seen 僕は落ちるのが怖い translated as "I am afraid to fall". Could it also be used to mean "I am afraid to fail".
Yes, it certainly could.
What this sentence means depends totally on the context.
"I am afraid to fall."
You are afraid of physically falling from a place of some height.
"I am afraid to fail."
You are taking an exam, auditioning for something, entering an election, etc.