Boskovich was born in Kolozsvár, Transylvania, Austria-Hungary (now Cluj-Napoca, Romania). The origin of his family and of his name was the town Boskovich in Moravia. In Cluj, Boskovich studied in a Jewish high school called "Culture" which accommodated both Neolog and Orthodox Jews. In 1920, Alexander joined the local Jewish organization "Hashomer" in which he was active for four years.