Look at the color of the metal, once it's been cleaned and polished. Brass has a more yellow coloring where copper or bronze have a reddish tint.
Check the type of dust it creates when drilled or shaved. Brass flakes off and bronze will be dust or string-like.
Pour hydrochloric acid onto the brass pull. If it turns a bright reddish pink, it is brass. It will look much like copper with the introduction of hydrochloric acid.
Find the era the pull was made. If it is labeled as antique, chances are it is brass. Newer pulls are made to look vintage, but are not actually aged.