Jewishness is determined through the Mother:


In a case where one of the parties is not marriageable according to Jewish Law the child born of their union has the status of the mother; i.e., a Jewish father and a non-Jewish mother-the child is considered a non-Jew.

Rabbi Shimon Bar Yochai, based it on the following passages in the Torah:

"You shall not intermarry with them [the non-Jews]; you shall not give your daughter to his son in marriage nor shall you take his daughter as a wife for your son. For -he [the non-Jewish father]- shall turn away your [grand]son from Me and they will serve other g-ds, and Hashem's anger shall be kindled against you and you will be quickly destroyed" (Devarim 7:3-4).

Intermarriage is prohibited both in the case of a Jewish boy or girl. The consequence of a grandchild being turned away from his Jewish faith is, however, mentioned only in regard to the non-Jewish father but not the non-Jewish mother.

The conclusion is that the child of a Jewish mother and a non-Jewish father is considered Jewish and the danger of his being diverted from his faith is relevant, while the child of a Jewish father and a non-Jewish mother is considered as non-Jewish and the consequence of being diverted from his faith is irrelevant.  Kiddushin 68b