Going all the way back to the start of the United States, there are a number of actions the government has taken that could be seen as attempts to increase public confidence in the banking system.
Perhaps the first such action wuld be the creation of the Bank of the United States. This was done in 1791. The basic idea was that a central bank would create a sound financial system that would inspire confidence in investors.
A similar desire was what drove the creation of the Federal Reserve system. This was done in 1913. It put all banks in the US under one regulatory system, thus improving confidence that the banks were well-regulated.
Finally, there was the creation of the FDIC. This was done as part of the New Deal. It gave people more confidence in the banking system because it guaranteed that they would not lose their money in the event of a bank failure.