What is the ruling on money received from tutoring the children of a bank employee, and should the tutoring continue or not?

If the bank where the children's father works is an Islamic bank, then there is no harm in dealing with him financially. However, if it is an interest-based (usurious) bank and all his work there is unlawful (haram), then it is not permissible to deal with him in any financial transaction, nor to accept his gifts or eat from his food. If his work is not forbidden, or if it is forbidden but he has another lawful income, then dealing with him is not forbidden unless it is known that the payment comes from unlawful sources. It is not obligatory to search and investigate the source of a Muslim's wealth; rather, the default is that it is lawful, and what is known to be unlawful should be avoided. If the nature of a person's work is unknown, then the default is the permissibility of dealing with him and teaching his children until something that forbids the interaction is proven.