I'm a total panicker in tight spaces and I worried about these same things. Here's what I found: I was fine on the way down. There's a ton of stairs that take you down. They wind around and around in a path. I was okay with them but they did make me wonder how I was going to feel at the bottom. Turns out, though, I was fine at the bottom. The caverns below are absolutely huge and I wandered around with the tour no problem. I was pondering the elevator ride up the entire time, though. It was ok but not great. They packed about ten of us in there, shoulder to shoulder. I closed my eyes and hid my face in my partner's chest and we were done.
I don't know what your terrible experience at the Holocaust Museum was - but if it was really really bad, I'll share this with you: the salt mine is interesting but it's a way longer tour than I thought it needed to be. I could have been in and out happily in 20 minutes and so if you think you just can't hack the elevator ride, look at a YouTube video of that main cathedral and don't worry about it further. It was the least interesting of the fabulous things I saw in and around Krakow.