An adverbial phrase is a group of words that functions as an adverb in a sentence. Many types of word groups can be used as adverbial phrases. Prepositional phrases are often used adverbially, and that's what we have in this sentence.
The prepositional phrase up the steep staircase is an adverbial phrase because it is modifying the verb <u>climbed</u>. It answers the question "Where did they climb?" It is an adverbial phrase of place.
