Is a worker in the support department of a company that sells software to various entities—including banks—sinful, bearing in mind that he does not know the nature of the work of the client whose problems he is solving?

If the products themselves are permissible, and the employee does not know the true nature of the client requesting support, then there is no harm in providing support, and he is not required to ask about it, especially if the majority of clients work in permissible fields, as the unknown is like the non-existent.