Banking systems will change anyway. It doesn’t matter if this is due to cryptocurrencies coming to power, or to some technological breakthroughs or disasters, but this will happen one way or another.
Banking systems are too unstable, these are financial bubbles, time bombs. People need more reliable systems or even abandon them.
Banking system will evolve but will never go away. Institution will take advantage of these innovation to have control over their citizen especially in finance sector. Imagine a government issuing a digital currency and implementing it to be the sole currency of the nation with its blockchain being private? They will track all the currency movement down to a single cent.
Blockchain will not change banking industry as much as we guys assume it would, I personally think they will only use it to make the data more transparent and easily transferable with less cost and higher speed but that won't come down to user and won't affect them at all.
Even if they somehow manage to make it faster which could help the customer the banking moves are quite fast already and if they find a way to make it cheaper it would just be less cost for them but same price for us, maybe whales would be able to transfer millions of dollars for cheap and that would help banks get whales but for the regular people it wouldn't change much at all. Banks don't care about the customer, they care about how much they make and blockchain won't change that too much for us.
Aside from that, I would say they will use it to spy on the people. Imagine a ledger where all the transaction is recorded and they have full control of it and each address/account have the people's real identity. That is kinda scary.