All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
All the followers of Judaism and Christianity that exist nowadays are among the People of the Book. It is true that they have changed, altered and corrupted the Books that were sent to their Prophets. But this does not exclude them from those known as the People of the Book. Indeed, they were known during the life of the Prophet Muhammad as the People of the Book and the rules pertaining to the latter were fully applied to them, although they did at that time change, corrupt and alter their Books. As for the marriage of the Muslim men from the contemporary Jews or Christians, it is permissible with two conditions:
1- The woman to whom someone proposes to marry, should not be among those of them who admit to be atheist, who do not believe in the faith of Judaism or Christianity; i.e. she must believe in God, and His Messengers (peace be upon them).
2- She must be chaste; i.e. she does not commit fornication or adultery. Because Almighty Allah described the one who could be married amongst them Saying (what means): {And [lawful in marriage are] chaste women from among the believers and chaste women from among those who were given the Scripture before you, when you have given them their due compensation, desiring chastity, not unlawful sexual intercourse or taking [secret] lovers..} [Quran 5:5]
For more benefit on marrying the women of the People of the Book, please refer to Fataawa 84159, 384922 and 397159.
Allah knows best.