It actually is possible to jump on stairs, with a couple restrictions. First, you need to stand in the middle of the step. Being too close to the next step or too close to the edge will cause it to fail. Second, you need to be facing to the right (move the mouse to aim the gun). 

The game's instructions don't actually mention spacebar to jump, so I didn't realize that jumping was possible at all until I read your comment. I spent a while trying to ascend the stairs to the right by phasing partially through them, and made it to the second-to-last stairs multiple times but could never get all the way up without knowing about jumping. But it turns out that the only thing up there was an ammo refill, so it was a waste of effort.