So, what does it give you? An immutable, distributed, public database that can be trusted despite not having trust in any particular participant.
Now, if you're a bank, why have you got offices with marble floors in the nicest neighborhoods? Why do your employees wear suits and ties? Why do you get a credit rating?
If you want security, that already exists. You can use encryption to identify people, and you can use it to hide sensitive information. You can use it over public channels, no problem. For instance this forum is on an encrypted https connection, and people can't sit on the route between me and YC to find out what I'm up to.
If you want a ledger, that's also been done. In fact you can use encryption to make sure there's never an unauthorized change to the ledger. And there's a whole load of ways that ledger can be scaled so that many people can use it at once.
Someone mentioned settlement of trades as a potential use case. But what is really to be gained against the way it is currently done? At the moment, you have a bunch of ops guys settling trades in the way it's been done for many years. If there's an error, you can call someone one the other side and fix it. Mostly, it's automatic, done by various cron jobs. What do we get from putting that in something similar to bitcoin?