ABN AMRO is shifting gears in its blockchain endeavor. The Amsterdam-based banking giant has announced a new service wherein individual client within non-bank organizations entrusted with client funds – think brokers, notaries, exchanges etc. – will directly connect with its clearing bank over a blockchain.
Link to CCN article
I think this is as good of place as any to get started with using the blockchain technology. With the technology being as new as it is and most people not being very comfortable with it, starting out with non-bank operations it's a good starting point.
I think other areas where this technology could be useful for banks would be in valuations. When banks order an appraisal for a loan, it has to go out to an Appraisal Management Company which has to then hire an appraisal to fulfill that order. Then the appraiser, upon completing the appraisal, sends it to the AMC which in turns sends it back to the bank. The appraisal can go back for corrections at any point in time if the AMC or the bank thinks it's not adequate. This process is very time consuming. The blockchain could solve so many problems in this aspect for the banks.