The standard view is that banking in a free market is inherently fragile, which makes deposit insurance necessary. In fact, some would argue that the concept of insurance needs to be extended to the so-called "shadow banking system." I think of Perry Mehrling and Gary Gorton as being in that camp.
The revisionist view is that deposit insurance is a case of the government concocting a solution to a problem that was created by government in the first place. That is, the U.S. banking system was unstable due to regulations that promoted small, local banks and inhibited the creation of diversified nationwide banks. Had banks been allowed to branch across state lines or had national bank holding companies been allowed to grow naturally, then (according to this argument) we would have seen few bank failures, even in the 1930's. Hence, there would be no need for deposit insurance.
There are several good reasons to argue against the traditional view that banking is naturally risky and therefore requires deposit insurance:
1. Maturity Mismatching is unnecessary. Japan, for instance, has historically adopted the view that lending of a certain type and duration should be matched with a liability similarly constructed. So instead of using flighty deposits to fund long-term projects, your deposits are funneled into assets with low risks and returns. Without crossing maturity lengths, the possibility of a bank run gets a lot smaller.
2. Absurd populist demands of the 19th century limited the number of branches a given bank could open. That resulted in a very geographically fragmented banking system prone to crisis every time agricultural yields fell in a given area. As a result, banks failed often and the entire business cycle was highly volatile.
Canada illustrates that the problem was exactly those banking restrictions, not banking in general. Canada's banking laws were far more permissive in terms of where banks could open branches, and the country developed a highly concentrated, well-regulated system of banks. By drawing on the deposits of an entire country, tiny shocks were not enough to force bank failure. In fact, the country barely experienced bank failures, and never saw banking panics -- even during the Great Depression. They did not even bother instituting deposit insurance until 1967. Canadian banks again outperformed American ones in the most recent crisis.
3. The past experience of deposit insurance before the 1930s was mixed. As Amar Bhide notes in A Call for Judgement, many of the state deposit plans performed quite badly through the Great Depression, frequently proving insufficient to cover all losses. The Indiana plan worked best by making other bank branches jointly responsible for the failure of a given bank. The national clearinghouse model also provided another vehicle for banks to collectively guarantee their deposits and prevent a panic, without requiring insurance. This sort of collective monitoring was not followed by other state plans, which performed poorly in sheltering depositors from bearing bank losses. Rather, by pushing some of costs of risky lending onto another party, they may have further encouraged bad lending and hastened banking crises.
The eventual adoption of FDIC insurance during the Great Depression drew on the disastrous experience of the standard issue insurance plans, as opposed to the successful mutual responsibility plans. It was was opposed by several interests -- including big banks, FDR, and the Treasury. Many of these actors were concerned by the poor performance of state insurance plans. However, federal insurance had the benefit of further entrenching the power of small banks, which would otherwise be a competitive disadvantage relative to their larger banking peers. We adopted the FDIC not as a part of a well-thought out plan to stem banking problems based on past evidence; but rather to satisfy the small bank lobby responsible for banking fragility in the first place.
4. The moral hazard aspects of deposit insurance have been tested internationally, and the results are fairly negative. A study by Aslι Demirgüç-Kunt and Edward J. Kane finds that countries with the highest coverage levels of deposit insurance are five times as likely to have a financial crisis as countries with low coverage limits. Better institutional quality (presumably leading to more prudential regulation) reduces the moral hazard caused by deposit insurance, but not entirely.
5. It's not clear that a banking crisis from one firm necessarily spreads to other, healthy, banks. The key test for this was the Great Depression, and my understanding is that studies point both ways on this.
At this point, FDIC insurance seems here to stay -- though it's worth pursuing policies to limit the amount of money covered by the insurance (say, at $100,000 per taxpayer, rather than $100,000 at each bank you have money at) or increasing bank co-insurance plans.
But this debate remains very relevant in thinking through the Shadow Banking crisis. One school of thought, best represented by Gary Gorton, feels that the problem there is similar to the issue in normal banking, and the solution lies in making repos -- the equivalent of deposits for Shadow Banks -- effectively riskless. Rather, the perspective above would point out the flaws in maturity mismatching generally, and would push against the institutional aspects of banking that make it more fragile by hiding risks and promoting moral hazard -- for instance, skewed salaries, the end of the partnership structure, and the bankruptcy treatment of derivatives.