Drywall – add framing to build a room under the basement staircase

It is not safe to simply cut a hole in the studs, and while the safe probably would transfer the load, you do not know that the safe was designed to transfer vertical loads, or how it will react. The consequences of a mistake here could be severe long term.

Therefore the proper course of action is to reframe the wall. Generally you would do this by adding a lintel (a RIGID horizontal beam that spans an opening typically over a window or door) at the top and bottom of the wall wide enough to transfer load to either side of the existing column, and then add new columns on either side of the existing post to carry that now transferred load down to the floor lintel and thus to the underlying support.

You could but why would you?

You are going to have to mud/tape your corners whether you take the drywall out or not. It will take you no more than 10 mins to dremel or knife out those two tiny sections of drywall.

Why you should:

1. You can see things easier.
2. You aren't relying on drywall for wall structure. I know this wall shouldn't be supporting anything but any weight at all will crush drywall.
3. Your nails or screws will not be binding to anything for 1/2-3/4 of an inch.
4. You will not be able to frame really tight unless you crush the drywall.

Points 3&4 could lead to a wobbly wall. Think of taking a big couch down in this basement. Corner of couch hit door frame hard. You don't want the top moving.