Identifying the problem
There are many different needs that both the banks and their customers have. A commercial bank will have different priorities from a retail bank, and they both will have different priorities from an investment bank. The blockchain technology is very well suited for settlement between parties. More complicated functionality can be either accomplished by a more focused technology like Open Transactions or more general smart contracts available for example on Ethereum. We will mainly focus on the settlement problem in this post, but will mention other issues as they fit in.
Is blockchain the way to go?
A blockchain is a very specialized tool. As discussed before, it is similar to a database, with a few notable differences. Performance-wise, blockchains might not be as efficient as centralized or distributed databases, but what they lack in performance they make up in other areas for some projects.
A blockchain enables accountability. A public blockchain like Bitcoin is a publicly verifiable, unalterable record of history. A permissioned blockchain may still be subject to forging of records if it is kept completely private, but if it is shared with an independent auditor or a digest of the records are anchored in projects like Factom, they can also be provably unalterable.
A blockchain is also a good way of reaching distributed consensus if the involved parties don't trust one another, say for international bank settlement between Russia and USA. If all parties run on the same protocol with conflict resolution, there is no doubt which transactions came through and in what order for example.
On the flip side, depending on the technology used, the blockchain might have lower transaction throughput than a dedicated, centralized server / database. At the moment the fastest ledgers update in 2-5 second intervals (in case of Ripple). To go faster, one would need to use ledgerless technology, such as Open Transactions.
Public versus permissioned blockchains
When we have identified which problem to solve and believe using the blockchain is the way to go, we should consider whether to use public or permissioned blockchains to help us achieve our goal. Pretty much everything that can be done on a public blockchain can be also achieved on a permissioned blockchain. Bitcoin and coloured coins act similarly to MultiChain. Ethereum can be approximated by Eris, and Ripple - by Hyperledger. Worst case scenario, most crypto projects can be forked from their open source repositories and modified to suit the particular needs.
Since the permissioned blockchain is handled by known, controlled servers, it can be pushed to achieve higher performance than a public blockchain by using higher-end hardware and network capabilities. Similarly, the more controlled environment is not at a whim of spammers, transaction fee fluctuation or any external forces.
That being said, a permissioned blockchain in many cases is a walled garden barring entry to a lot of possible innovators. While this can be useful for protecting sensitive information, you can also miss out on the network effect from using the public blockchain. It's just like the internet and the intra-net - both have their uses and drawbacks.
For a lot of applications, it might make sense to either be completely on a public blockchain, or at the very least operate on both the public and permissioned blockchains at the same time. For example, if SEPA operated its own permissioned blockchain to offer settlement between all the European banks, it could also offer similar services on a public blockchain for lower-frequency transactions and perhaps a bridge to connect between the blockchains. This way if someone decided to build, say, a settlement corridor between Europe and America, they could use the already available public blockchain without having to apply for a banking license to get access to the premissioned one. This would enable more innovation to take place on low-performance environment while keeping the core network performing very efficiently for high-volume transactions.
Just to note - a public blockchain does not mean everyone would be able to use the solution indiscriminately. Pretty much every blockchain offers some way of controlling who can access various currencies issued on it, either through the use of white- and blacklists or through multisignature. This means that the banks or governments can still follow with KYC and AML requirements, even if the blockchain is public for viewing.
Picking the right approach
As with most situations, there are many ways to approach a problem on a blockchain. For example, in order to do international settlement, one can issue fiat-denominated currencies, copy an existing FX market and use that for currency conversion, use an intermediary cryptocurrency like Bitcoin (an approach used by Abra for example), or perhaps create a smart contract to handle the trades. The number and kind of options will depend on your problem at hand and it's hard to generalize this point.