Rabbi Blumenkrantz was born in Mandatory Palestine to parents of Ashkenazi European extraction, Rabbi Chaim Menachem Bentzion and Rebbetzin Devorah.
Avrohom began his education in Bogotá, but came to New York City as a teenager to study at Mesivtha Tifereth Jerusalem under the famed Rabbi Moshe Feinstein, with whom Rabbi Blumenkrantz maintained a close relationship until Rabbi Feinstein's death in 1986.