Firstly, Allah Almighty has only permitted marriage outside of Islam if it is with women from the People of the Book:
And lawful for you are chaste believing women and chaste women from among those who were given the Book before you, when you give them their dowries, contracting valid marriage and not committing fornication nor taking secret paramours.
(Sūrah al-Mai’idah 5:6)
However, this does not mean that marriage with all women from the People of the Book is permitted. Allah Almighty has forbidden marriage with women who commit shirk:
And marry not idolatrous women until they believe.
(Surah al-Baqarah, 2:222)
Describing those of the People of the Book who commit shirk, Allah Almighty says:
They are surely disbelievers who say, ‘Allah is the third of three;’ there is no God but the One God.
(Sūrah al-Mai’idah, 5:74)
Allah has not taken unto Himself any son, nor is there any other God along with Him.
(Surah al-Mu’minun, 23:92)
Marriage with a woman from the People of the Book is permissible if her doctrine is the unity of God. Allah Almighty says:
They are not all alike. Among the People of the Book there is a party who stand by their covenant; they recite the word of Allah in the hours of night and prostrate themselves before Him. They believe in Allah and the Last Day.
(Surah Al-e-‘Imran, 3:114–115)
However, Hadrat Khalifatul-Masih IV rta explained that if she holds doctrines that are shirk, then marriage with her is not permitted even if she is from among the People of the Book (Question and Answer Session, May 24 1997).
Now remains the question of whether this permission results in wives exercising more influence on the religion of the children than the husband. This is a case where correlation does not imply causation. Unfortunately, many men who marry non-Muslim women are not strong in their religion. The true purpose of marrying non-Muslim women is to spread the message of Islam. However, many of these relationships start in a way that is un-Islamic, for example through dating. It is not surprising when the husband’s weakness of faith and his wife’s lack of faith result in their subsequent children not following Islam. It is not the permission to marry non-Muslim women that causes children of such marriages to drift away from Islam. Rather, it is the inappropriate use of this permission by Muslim men who are weak in their faith that is causing the children of such marriages to drift away from Islam.
Although a non-Muslim wife does not have to accept the spiritual teachings of Islam, when getting married, a Muslim man should discuss the role that Islamic social teachings will have on their life together. The non-Muslim wife should be aware that although she does not have to convert to Islam, she is still coming into a Muslim household wherein the husband is the head of the household (Surah an-Nisa’, 4:35). Moreover, she must also be aware that the children are to be raised Muslims. The Holy Prophet sas said:
A man is a guardian of his family and is responsible for them; a woman is a guardian of her husband’s house and children and is responsible for them.
(Sahih al-Bukhari, Book 67, Hadith 134)
Ordinarily, when people are getting married, they discuss things that are important to them that could potentially create conflict in the future. If Islam is important to the man, he will make this point clear. If the husband gives priority to Islam, the issue of his wife defying his faith and his children turning away from Islam should not arise.