Is it prescribed for a Muslim to sit with his family if this sitting mitigates their reprehensible acts (for example, if cursing and slandering turn into backbiting, or if watching video clips turns into TV series), knowing that he avoids their reprehensible acts as much as possible?

It is not permissible to sit with people who commit sins or introduce innovations, even if you reduce your interactions with them due to their disobedience. This is because preserving one's own faith takes precedence over attempting to reform people at the expense of one's own integrity. If they persist in a particular transgression and do not respond to you, then you must part ways with them.

Allah Almighty says: "And He has already revealed to you in the Book that when you hear the verses of Allah being denied and ridiculed, then do not sit with them until they engage in another conversation. Indeed, in that case, you would be like them. Indeed, Allah will gather the hypocrites and disbelievers in Hell all together" (An-Nisa: 140). Thus, being content with sin is a sin itself, and whoever sits with people of sin and does not condemn them is like them in their burden. If one cannot condemn them, then one must leave their presence.