climb: to move or go up (something) using your feet and often your hands
climb: to ascend, go up, or get to the top of, especially by the use of the hands and feet or feet alone or by continuous or strenuous
effort: to climb a rope; to climb the stairs; to climb a mountain
Because climb when used without an adverb or prepositional phrase (e.g., over, over the rocks, down, down the stairs) implies "up", one should leave out "up" in this case.
So, either of the following would be equally correct:
I climbed the stairs. (Not "I climbed up the stairs.")
I walked up the stairs.