Wa `alaykum as-Salamu wa rahmatullahi wa barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear brother, it is indeed very appreciated from you to ask this question, as it is each Muslim’s duty to fully observe his/her income making sure that it is all gained from lawful (halal) sources.
Working for a bank is permissible if one doesn’t take part in interest-based dealings.
In his response to your question, Prof. Dr. Monzer Kahf, Professor of Islamic Finance and Economics at Qatar Faculty of Islamic Studies, states:
Working in a bank is permissible (this does not mean it is one of the best jobs a Muslim may have).
However, signing any interest-based transaction or writing it or reviewing its correctness for the purpose of signing it or approving it and the like are all haram and a Muslim should avoid doing it.
It is under the curse of Allah and His Messenger as stated in a very authentic hadith. Whenever what you do is haram, its income is not permissible too.
Allah Almighty knows best.