Is it permissible to boycott a brother with the intention of disciplining and training him for visiting suspicious websites, and how long can this boycott continue?

If your brother visits suspicious websites, this is a transgression on his part, and it is permissible to abandon him if gentle and kind advice and guidance do not benefit him. Abandonment is permissible if it is thought to benefit the one being abandoned; otherwise, it is better to leave him if it will only increase his stubbornness and persistence in evil.