Is the husband sinful for preventing his wife, whom he divorced due to her betrayal, from seeing her children, in order to protect the children from her environment?

The wife has the right to see her children, and the husband does not have the right to prevent her from doing so. He would be sinful for such an act, due to the قول of the jurists: "When a child is with one of the parents, the other parent is not to be prevented from seeing them or caring for them," and their قول: "Neither of them should be prevented from visiting them when they are with the other." What you mentioned about humiliation or protecting the children does not justify preventing her from visiting them. If a dispute arises, the matter should be referred to the Sharia (definition) court.