During the Great Depression, there were runs on banks. Even though the Fed was printing money, the runs on banks led to, in effect, a contraction of the money supply, due to a reversal in the fractional reserve lending process. Out of this, came the FDIC, another Roosevelt government program that is still causing massive damage to this day.
It sounds scary to most people today when they consider a world with no FDIC. After all, how are they supposed to be assured that their money is safe? First, it should be pointed out that there are no guarantees in life no matter what. There are a million different ways that you could lose your money, with or without the FDIC. Second, with no FDIC, consumers would actually pay attention to the solvency of banks. Now you many think that you don't have the time or knowledge to research banks, and that would be the case for most people. But like anything else in our economy, it would be done by the whole market. You are not a mechanic and yet you manage to buy a car. You are not a farmer or food inspector and yet you manage to buy safe food. It would be the same with banking.
The FDIC has created huge moral hazard. Banks have taken on far more risk than they ever would without this insurance. In addition, it is the Federal Reserve with its power to create money out of thin air that allows the FDIC to exist. Without the Fed, the FDIC would not be able to guarantee deposits. The FDIC would be broke in no time.
This whole system creates a huge problem. It is one of the reasons for the panic that happened 2 years ago. It was one of the reasons for the huge bailout. If there had been no bailout, there would have been bankrupted banks with depositors losing their money. Then the FDIC would have stepped in and gone broke. Then the Fed would have bailed out the FDIC. This chain of events would have been better, because at least the bailouts would have just gone to depositors and would not have propped up failing banks.
In a libertarian world, there would be no Fed and no FDIC. The best case scenario now is to phase out the FDIC. Everyone would be surprised by the changed behavior of the banks. You might have to start paying a fee to have a checking account with a bank. But at least we wouldn't have huge bailouts and a constantly depreciating currency.
The FDIC is a tough issue for libertarians. Should depositors be stiffed? How do we go about eliminating the FDIC without causing a major crisis? Do we abolish the Fed first or the FDIC? Or do we abolish them in tandem? There are a lot of interesting questions. One thing for sure though is that there is no place for the Fed and the FDIC in a free market.