It is. Note, though, that it means "I am walking downstairs", "I am currently at the stairs, going down". It makes sense in some contexts, it doesn't add up in some others. For example, imagine you are in a large railway station where you are supposed to meet your friend. The friend doesn't seem to be there, so you make a call and ask "Where are you?". Then the answer "Ты в главном зале? Я иду вниз по лестнице" is logical and quite natural (i.e. "look at the main staircase. I'm right there, walking down").
However, usually it isn't important and we don't even remember walking downstairs to exit the building/ going upstair to the office. You mostly just use "спускаться"/"подниматься" without mentioning the stairs/elevators, let alone "иду вверх"/"иду вниз".