There's no evidence at all in the books that any of the staircases are open like that, and with between 300 and 1,000 (depending on whether you look at the math or JKR's statements) 11 to 17-year-olds running around there 10 months of the year, it'd be a fairly bad idea. They'd constantly be trying to push each other off or throw others' possessions down, etc. However, it makes for a much more dramatic shot than a simple large marble staircase and so they show it as that in the movie.
That said, there would have to be, at the least, a fairly strong, permanent cushioning charm at the bottom (and potentially various spots along the way). However, I bet the children would quickly figure out that it's great fun to jump off the highest staircase onto said charm, so I'd imagine there's also a wall-like charm to prevent people (and their possessions) from falling, being thrown, or themselves jumping over or through the railing.