Having a block chain doesn't mean that you have to give access worldwide to anyone and everyone. Banks could provide blockchain access to only computers that cleared their firewall, and are API authorized. Think ach except with a black chain as th back end.
Trust isn't a binary relationship. Computer security is hard. I trust people to try their best to not get hacked or not to get kidnapped and tortured, but alas it sometimes happens. I'm surprised no-one has mentioned that the only reason to use a block chain is to achieve byzantine fault tolerance. Casper is trying to solve much the same thing that has already been solved by Tendermint but with different trade-offs (CAP theorem trade-offs).
Well banks don't necessarily trust their international partners for example totally. Smaller banks fo instance have a lot of trouble sending unsecured wires. With the block chain it would be possible to have more trust on the amount of funds another bank has available.
