I downloaded this apartment lot a while ago and I really like it, but it has some spiral stairs placed in such a way that sims can't climb them. I'm assuming that the person who made the lot has a mod that allows these stairs to be traversed that I don't have. Where can I find it?

I'm not sure, but maybe this mod can help: https://mustluvcatz-reloaded.tumblr...ted-all-3-files

No, I don't think that's related.

Maybe something like this

That only works for one-tile objects, and probably not build-mode objects like stairs anyway.

I always layer spiral stairs with no problems, saves space in tiny homes. Although, judging by the picture, the problem is 2 sets of spirals with no floor between them. Each set needs a 1 tile square to step off, then head up the next set. Hard to tell by the small image. Not sure how to find it, but I did post an image somewhere on MTS showing how to stack spirals from one level to the next, you need 1 tile at the top of the first set where they would exit and the 2nd set uses the same tile to go up another level. You would probably need move objects cheat too.

The two sets of stairs are not related (they are not even accessible one from the other), and no one has a problem using the shadowed out stairs, that I know of. I'm talking about only the stairs in the foreground. A sim that is standing in front of the stairs can climb them, and a sim that is at the top of the stairs can descend to stand in front of them, but a sim standing in front of the stairs cannot go anywhere else on the same level, and a sim elsewhere on the same level cannot move to stand in front of the stairs.

It is possible that the lot wasn't playtested before uploading and you'll have to move the stairs.

Have you tried rotating the stairs to see if you can find a better placement? I'm guessing sims need two grid squares in front of the stairs and at the top to be able to access them properly. You could also try removing some of the wall. If all else fails, there could be space for regular stairs, but it's hard to tell from the picture.

Is there a safe way to edit the apartment common area without moving my sim out of the apartment?

No, unfortunately. I tried it and it led to problems with the other apartments. A new room I built people kept trying to walk through. You'll have to move them out and in again.