An adverb is a function word which acts as a modifier of an adjective, a verb, another adverb or a word group. In this sentence, the adverb phrase, which is also a prepositional phrase, is "with either hand." It starts with a preposition (with), followed by the adjective "either" and ends with the noun "hand."
In this sentence, the word it modifies is the noun "basketball" as it expresses the medium through which the ball can be shot.