[Azaelia of Willowbottom]

Good question. Perhaps someone more experienced than I can provide a more satisfactory answer, but here's my shot at it.

Perhaps the stairs had eroded over time...They were built a very long time before LOTR took place. They can't have been easy to carve in the first place: going almost straight upwards. Maybe they were only able to carve thin steps.

A simpler answer would be that it makes for dramatic movie shots. The sense of vertigo during those scenes in the movie was pretty strong, and I doubt that it would have been quite so strong had the stairs been shown as larger.
I can't remember exactly how they were described in the book, but I *think* that the stairs being so thin may have been a PJ addition.