The scenario is that you have a thick stone-layer aquifer. This is my quick and safe (and cheap in terms of resources) method for piercing it with a 3x3 stairwell. Recommended workforce: as many miner/engraver/mechanic/pump operators as you can spare. Two or three are quite sufficient. Also components for a screwpump or two and some grates.
Start with a setup like this, pump to the west:
▓▓▓▓▓
▓▓▓.>.▓
▓#%%≈▓▓
▓▓▓...▓
▓▓▓▓▓
The watery open tile should have an up-down staircase dug beneath it and the top channeled open, so now let's go down a Z, which should look like this:
▓▓▓
▓X▓
▓▓▓
Start pumping and dig up-down stairs under the down-stairs:
▓▓▓
▓X▓
▓X▓
▓▓▓
Now dig up-down stairs under the CENTER tile. If there's another aquifer layer it'll be your drain and you can stop the pump; if not, it means you've found the first non-aquifer layer and just have to keep the pump running. Now smooth the whole thing, then dig more up-down stairs and smooth around them one by one, until your staircase looks like this:
╔═══╗
║XXX║
╠OXO╣
║XXX║
╚═══╝
Channel out center stairs, dig out tiles beneath the pump, smooth all leaky walls, channel the new side-drain,
╔═══╗
▓═╝XXX║
▓▼..≈O╣
▓═╗XXX║
╚═══╝
install grate, smooth out remaining tiles, move the pump,
╔═══╗
╔═╝XXX║
║#%%≈O╣
╚═╗XXX║
╚═══╝
and repeat as long as necessary.
Extra bonus: you can build a well on top of the last side-drain aquifer, where it is easily accessible.