Is earning from the Internet considered permissible (halal) or forbidden (haram), knowing that it is the sole source of income for a Muslim family?

The Internet is used for both lawful and unlawful purposes. If it is possible to control the premises and prevent the use of the Internet for unlawful purposes, then the resulting earnings are lawful. However, if the owner of the premises neglects to admonish his customers and prevent them from engaging in unlawful acts, he is sinful, and the earnings derived therefrom are unlawful.