Is the husband sinful if his wife sits with his family to watch television in their home, knowing that advising them is futile, and is it obligatory for him to definitively prevent her from sitting with them?

If the wife limits her television viewing to what is permissible, such as news programs presented by women or religious programs free from mixing (of genders) and forbidden things, then there is no harm in that. However, if she watches indecent programs or mingles with non-mahram men, then the husband must prevent her, based on the words of Allah Almighty: "O you who have believed, protect yourselves and your families from a Fire whose fuel is people and stones."