I remember having the same problem 10 years ago:

Soon after the start of Fallout 1 I get stuck at a broken elevator shaft in vault 15.
I "sucsessfully use the rope on the elevator shaft" but I can't climb down the rope.
What am I doing wrong?

Typically you need to click on the rope, there should be a hand icon.

No matter how many times I click on the rope hanging in the shaft, I cannot go down the shaft.
What am I doing wrong?

I still can't climb down the rope.
Every time a click on the rope with the" hand icon" it changes from "the rope is hanging down the shaft" to "the rope is hanging above the shaft"

PLease help....

Click the shaft instead of the rope.... thats about as far ive got with my testing of the game, and as far as i remember i clicked the shaft... that game sure has some control flaws

Clicking on the shaft just lets me change the rope back an forth from "hanging down the shaft" to "hanging up the shaft"

What does it take to CLIMB down the shaft using the rope?

Are you sure that you aren't climbing down the shaft already? Seems to me that if it says "hanging down the shaft" and you click on it then you climbed down the shaft, because after that it says "hanging up the shaft". Check your surroundings.

BRILLIANT!!!!
Boy, do i feel stupid.
The worst part is, now I remember feeiling this stupid ten years ago when I first got stuck.

Thank You very much. You made my day.