Husband’s responsibility towards his wife and children
Question

With regard to the husband being “the shepherd of his family and responsible for his flock,” what is the level of knowledge and religious commitment that is required of him? If for example the wife or children do something that is forbidden in sharee’ah, will the husband be sinning and deserve the punishment of Allaah for failing to fulfil the trust and not advising them before they did it?
Praise be to Allah.For information on the attributes of the righteous husband, please see the answers to questions no. 5202 and 6942 .
Secondly:
The man is a “shepherd” in his house and he is responsible for his flock, as it is narrated in the saheeh report from the Prophet (peace and blessings of Allah be upon him). He is responsible for teaching and educating his wife and children. If a person falls short in that and his wife or children fall into sin, he will also be sinning, because he is the reason why they were not educated and taught. If he does not fall short in that but some of his family fall into sin, then he will not be sinning, but he has to remind them and exhort them after they fall into sin, so that they will give up the unIslamic things that they have fallen into.
Shaykh Saalih al-Fawzan (may Allah preserve him) said:
Teaching of children begins when they reach the age of discernment, so he should begin teaching them about religion, because the Prophet (peace and blessings of Allah be upon him) said: “Tell your children to pray when they are seven years old and smack them if they do not pray when they are ten, and separate them in their beds.” Narrated by Abu Dawood, and it is saheeh. When the child reaches the age of discernment, then his father is enjoined to teach him and train him in that which is good, by teaching him Quran and whatever he can of hadeeth, and teaching him the shar’i rulings that are appropriate to a child of this age, so he should teach him how to do wudoo’, how to pray, and the dhikrs for going to sleep and waking up, and for eating and drinking, because when he reaches the age of discernment, he can understand what he is being told to do and what not to do. Similarly he should be told not to do things that are inappropriate, and it should be explained to him that it is not permitted to do these things, such as lying, gossiping etc, so that he will be trained in what is good and to give up what is evil from an early age. This is a very important matter which some people neglect with regard to their children.
Many people do not pay attention to their children and they do not educate them properly; they leave them alone and do not teach them to pray, or teach them what is good, so they grow up ignorant and doing things that are no good, mixing with evil people and loitering in the streets and neglecting their studies, and other harmful things with which many Muslim youth grow up because their parents are neglecting them. But they are responsible for them because Allah has placed the responsibility for their children on their shoulders. The Prophet (peace and blessings of Allah be upon him) said: “Tell your children to pray when they are seven years old and smack them if they do not pray when they are ten, and separate them in their beds.” This is a command that is addressed to parents, so the one who does not tell his children to pray is disobeying the Prophet (peace and blessings of Allah be upon him) and doing something haraam, and is failing to do a duty that has been enjoined upon him by the Messenger of Allah (peace and blessings of Allah be upon him).
The Prophet (peace and blessings of Allah be upon him) said: “Each of you is a shepherd and is responsible for his flock.” Narrated by al-Bukhaari and Muslim. But some parents, unfortunately, are busy with worldly affairs and do not pay attention to their children, and they do not give them any of their time, rather they spend all their time in worldly matters. This is a grave danger that has become widespread in the Muslim world, and has led to poor upbringing of children, so they do not succeed in religious or worldly terms. There is no power and no strength except with Allah, the Most High, the Almighty."(Al-Muntaqa min Fatawa al-Shaykh al-Fawzaan  5/297, 298, question no. 421)
And Allah knows best.