While the games faithfully recreate many Hogwarts areas, the moving staircases are not implemented as interactive elements. You can't climb them like you would in the books and movies. This is likely due to technical limitations of the game at the time.
You might find staircases leading to different floors in certain areas of the game, but they won't be the "moving" variety.
To experience the magical staircases in a Lego Harry Potter game, you would need to play later installments like Lego Harry Potter Years 5-7, where they are included as interactive elements.