The ascent of the Stairs of Cirith Ungol by Frodo, Sam, and Gollum is an important scene in The Two Towers. There are a bunch of other significant staircase scenes in the trilogy.
Robert Louis Stevenson’s Kidnapped, made into a number of TV and film adaptations has an important sequence involving stairs. I remember seeing one on the telly when I was young(sometime in the 60s), which made a vivid impression. The young hero, staying at his wicked uncle’s half built half derelict, house has been coerced into going upstairs in a disused part of the building. Just in time for him lightening reveals the stairs ahead are broken. Our TV at the time was black and white so I don’t even know if this was originally in colour or b and w and can’t remember whether it was a film or made for TV so I doubt I can find out which version this was.
Mystery Men has a cliche-busting fight on a staircase and balcony, with a truly hilarious punchline.
Moulin Rouge has one of its most powerful dramatic moments when Nicole Kidman’s character slowly descends the stage steps, in an elaborate costume, to submit to the Duke’s demands.
A hokey David Niven film called A Matter of Life and Death featured an escalator between this world and the afterlife. The movie was sold in the US as Stairway to Heaven.
There are a lot of stairs in GWTW. Scarlett slapping Prissy, Scarlett falling down the stairs and having a miscarriage, Scarlett coming down the stairs after the Yankee deserter tries to steal from them, the scene where Mammy tells Melanie why Rhett won’t let them have Bonnie’s funeral was filmed as they were walking up the stairs, etc.
Odd that I can’t find it on YouTube, but the staircase from The Little Mermaid. It’s notable not for its role in the story, but because it was one of the first major uses of computer animation in a movie.