Translation:Take the stairs to the second floor.
I think that лестнице is dative ("leading to [somewhere]"). It could be prepositional, but I don't believe it fits the definitions for prepositional objects of по.
In either case, лестнице is singular. Although it is translated as "stairs", it can also be translated as "staircase", which means "the set of stairs leading from one level to another". In English, "staircase" is also singular, though "stairs" is plural.