Banks would not be trusted without the State

The effect of deposit insurance is that people are not concerned with the solvency of their bank and are not really worried about a bank run. They feel protected from any loss and do not consider it prudent to take preventative measures to take their savings out. This is possible, to provide the insurance, because paper money may be issued in any quantity and there is no (material) limit on how many notes can exist...

Without deposit insurance it would be more difficult for a bank to lend imprudently because depositors would be wary and look out for any bank making loans and subsequently, withdraw their funds. It would not be possible for the bank to make any loans, because people would want their money available on demand, for everyone. A rumour that a bank is loaning money might be disastrous for the bank if, as a result, they lose all their customers.

Deposit insurance enables banks to keep their deposits even though they have loaned the money to someone else. Without deposit insurance, and without the enforcement of the contract to take good care of the deposits, people would not trust to use the banks. It is because of this contract that people are willing to 'trust' the banks, in fact this is the reverse of trust, if we rely on the threat of violence to provide confidence.

If there is no enforcement of contracts, as there should not be, then no bank would exist; the reputation would not be sufficient for people to think it a good idea to hand over their belongings.