It's a distributed trust network, right? Why would banks that survive on trust want that distributed?
Well there's two parts to it, one is the "chain" property where like git's commits it's not possible to edit one transaction later and have it go unnoticed. You can run independent background audits that confirm that this blockchain state corresponds to these transactions and account balances. It's a lot more difficult than adding one fraudulent transaction by itself, like that somebody deposited cash in your account when they never did. Obviously if you can add "genuine" transaction to the chain that's different, but they can be validated in the process.
The other part is inter-bank transactions where it's essential that everybody agrees on the state of affairs. I wouldn't use the "proof of work" but rather signatures of trusted parties, one party one vote. If 100 banks get an inter-bank ledger, 98 of 100 agree on the block chain all the alarms should go off in the last two banks. With signing and countersigning it's pretty hard to go back on anything as 100 banks have digitially signed that they saw your bank digitally sign that this block chain is correct. Because it's harder than you think to find one trusted master to rule them all, both domestically and internationally. Everybody wants to do their own verification which is exactly what block chains provides.