I'm pretty sure banks will also join the crypto currency world if fiat become obsolete (which i doubt will). They will do the same thing they do in banking like loans, savings, investments and exchanges. Maybe they'll also have their own coin and or wallet that is highly theirs. I don't doubt that even now, banks have stakes in Crypto currency.
Eventually, when the time comes where the Cryptocurrenciea have become more popular and more widely used globally, then banks have no choice but to cope up with what the market demands or else they have to shut down. In a digital age, I am sure that there is a possibility that fiat money will somehow become obsolete.