There is an option to remove stairs and constructions on the digging menu - but your dwarves have to be able to access the tile they're removing, which makes removing a stairwell in a hole rather difficult.
You could dig the hole a little wider and deeper and try to use it as a well - or just cover over the top with constructed floors so no one goes down there.

I'll note that if the first place you dug had water, almost anywhere probably will! It sounds like you're hitting an aquifer, so to get through it you'll need to dig down, quickly smooth/construct walls in the tiles next to your stairs, then dig down again. Smoothed or constructed tiles won't leak water and most aquifers are 'light' aquifers which leak plenty slow enough to pull this off.

Stair technology can be pretty complicated.
Build some floor over it, it will cut the path and keep anything from coming up.

Keep in mind dug stairs are slightly different from "built" stairs. Both can be removed, but if you don't want to or can't access the lower area you can just build a floor tile on the upper area and it will completely cut the path in both cases.