All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Islamicly, there is no harm in this as far as we know. On the contrary, if a non-Muslim opens a bank account in an Islamic bank, this is better than opening an account in banks which deal with interest and usury. If one does so, he is reducing the number of people who are contributing to sin and enmity.
On this occasion, we refer the questioner to the following Fataawa (plural of Fatwa): 81979, 81396, 83990, 88187, 88276 and 88906.
Allaah Knows best.