An appointment letter or order is nothing more than the confirmation of the offer of employment or the selection of the candidate for the job by the employer. Therefore, there is no compulsion, either statutory or contractual, on the part of the candidate to accept it. They can reject it without citing any reason or simply be non-responsive.
However, the reason you expressed for rejecting the offer based on your family's views seems a bit unusual. Of course, they have every right to have such a perception about private banks, and you do have the right to abide by the wishes of your family. Anyway, I don't think that you have been selected for an appointment in a private bank for any top posts like chairman, general manager, or financial controller, and the like. What about public sector banks, public enterprises, and state and central government departments? Are they above corruption and malpractices? Perhaps the elderly members of your family may have some concern for job security, which can be genuine. However, when you are highly employable, such a concern automatically pales into insignificance. Therefore, think on your own, consult your peers already in service, and try to convince your family, if possible, before simply rejecting the appointment.