All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
In principle, it is forbidden for a Muslim to open an account in a usurious bank because of the following:
First: If he accepts usury then he is an eater of usury.
Second: If he doesn`t accept usury, then he knows that usurious transactions rest on such accounts and funds. The evidence on this is that Allah The Almighty Says {What means}: "Help ye one another in righteousness and piety, but help ye not one another in sin and rancour: fear Allah. for Allah is strict in punishment.” {Al-Maidah/2}. Moreover, on the authority of Jabir: “The Messenger of Allah (PBUH) cursed the one who accepts Ar-Riba (the usury), the one who pays it, the one who records it, and the two persons who stand witness to it, and he said: ”All of them are equally sinful.” {Muslim}.
If the usurious bank grants a prize to its client in return for his having an account there, then that prize comes from what is unlawful, so it is forbidden for him to benefit from it. Rather, he should get rid of this ill-gotten money through giving it to the poor and needy and in favor of public interests. And Allah The Almighty Knows Best.