My cousin works in a bank but he works in HR.
As a principle it should be remembered that the permissibility or impermissibility of employment in a certain firm or company must be lawful, meaning if the work entails something that is prohibited in Shariah, such as serving alcohol, calculating interest transactions, and so on, then this type of work would be unlawful. Now, employment at a bank that entails direct involvement with interest based transactions such as the job of a manager, cashier, clerk, and so on, would be unlawful.
Sayyiduna Abd Allah ibn Mas’ud narrates that the Messenger of Allah (Allah bless him and give him peace) cursed the one who accepted usury, the one who paid it, the witness to it, and the one who recorded it.
-Sunan Abu Dawud: Book 22, Number 3327
In another Hadith it is evident how the Prophet Muhammad invoked curse on the receiver of interest, the payer of interest, the scribe, and the two witnesses.
Jabir said that Allah's Messenger (ﷺ) cursed the accepter of interest and its payer, and one who records it, and the two witnesses, and he said: They are all equal.
-Sahih Muslim 1598
Hence, the warning of Prophet Muhammad is very severe and quite clear. So, the one who is directly involved with interest transactions in any way, shape, or form, such as writing or calculating the transaction is conducting an act of abomination in the eyes of God. Therefore, one must avoid accepting a job at a bank, which consists of being involved in interest transactions and dealings. It is unlawful to help and assist others in their interest-based dealings as evident by the traditions and sayings of the Prophet Muhammad. As such, one must decide between the choice of temporary worldly gains or the life of eternal peace and joy in return to God upon the next life. For one must always understand that the joys of this world is only temporary and the next life that is of permanent settlement, so do not be deceived into the delusions of this worldly life.
O my people, this worldly life is only [temporary] enjoyment, and indeed, the Hereafter - that is the home of [permanent] settlement.
Know that the life of this world is but amusement and diversion and adornment and boasting to one another and competition in increase of wealth and children - like the example of a rain whose [resulting] plant growth pleases the tillers; then it dries and you see it turned yellow; then it becomes [scattered] debris. And in the Hereafter is severe punishment and forgiveness from Allah and approval. And what is the worldly life except the enjoyment of delusion.