Anthony is Jewish (JKR:Tw).
Anthony is distantly related to Queenie Goldstein and Porpentina Scamander (JKR:Tw).
Anthony is the English form of the Roman last name Antonius. The name became popular because of Saint Anthony the Great and Saint Anthony of Padua. Goldstein is a German-Jewish last name combining the words "gold" and "stone."
According to the notes shown in the 2001 Harry Potter and Me BBC special, Anthony was a half-blood. However, they also show that he was a Hufflepuff, which isn't true in the books.