In English, some people say "have a fear of", but the far more common expression for "avoir peur de" is "to be afraid of". Even though it literally looks like we would have to work "have fear" into the English, you are encouraged to use natural English. Consider these other expressions using "avoir".
- J'ai vingt ans = I am twenty (years old) -- not "I have twenty years".
- J'ai faim = I am hungry -- not "I have hunger"
- J'ai froid = I am cold -- not "I have cold"
- J'ai besoin d'une pause = I need a break -- not "I have need of a break"
- J'ai envie d'aller au cinéma = I feel like going to a movie -- not "I have a desire to go to a movie"
- Tu as raison = you're right -- not "you have reason"
Check this link for more expressions using "avoir": https://www.thoughtco.com/french-expressions-with-avoir-1368646
Certainly it's natural to see it that way if we're still looking at "j'ai peur de...." in a literal way. It is not unheard of, in certain contexts, to say "I have a fear of..." (the article is required in that sense). But we come to understand that the natural counterpart of the French expression is "I'm afraid of ...." Open this link in a browser: http://www.wordreference.com/fren/avoir%20peur%20de
Just "falling" will suffice. Choose the natural English expression which is a counterpart to the French "J'ai peur de tomber" = I am afraid of falling. No "down" is necessary. http://www.wordreference.com/fren/avoir%20peur%20de