Bitcoin _can_ solve this problem because it is totally distributed and doesn't depend on traditional banks which means transferring value directly between individuals and organization becomes much simpler.
Currency (at least metal coins) was, at first, distributed and did not depend on banks for transfer. Banks came about later on, as an additional help in some business and private money handling. Currently banks are pretty much necessary for two reasons: network effect -- everybody has bank account and expects you to interface with it, and better loan rates -- banks can give loans at lower percent than small time lenders, thanks to pooling of risk and economies of scale of processing.
It is expected some Bitcoin users (be it private or institutional) will become quite bank-alike in some future (as per various posts on blogs and HN in discussions of Bitcoin).
In short, Bitcoin doesn't remove banks -- neither need for them, nor ability to operate them. They just haven't moved into this space yet.
EDIT:
first episode of ``The Ascent of Money'' [1], `Dreams of avarice', explains how banking, as we know it, came to be.
Which is the same as mailing cash. It's possible to do that without a bank as well.
Since the Bitcoin protocol allows third-parties to hold user's wallets, it's entirely possible to create a bank. In fact, the protocol seems to encourage the aggregation of wallets.
A better example would be paying for something with cash. You're taking money, and giving it to someone else(presumably in exchange for a good or service).
The point is that you're moving your currency(BTC, USD) from you to another party without any intermediary interfering with the transaction. If you mail $1 to someone, that's effectively the same thing as paying them with 1 BTC.
EDIT: I think the people who are saying that Bitcoin cannot have banks are conflating payment processors with banks. A bank is a place where you go to put your money, and they use that money to make loans, which gets them more money through payments on loan interest. They incentivize this behavior(giving them money) by giving out interest on the amount in the accounts.
Payment processors are companies(i.e. Visa, MasterCard, etc.) that let you take your money from an account, and give it to someone else. Banks may offer these in the form of checking accounts, but that is not their intended purpose. These, I agree, are definitely not needed with the Bitcoin system, because that's what the entire protocol handles.
