Bank Runs During the 1930s Why were there numerous banks run right after the stock market crash of 1929? Why would a bank fail because the stock market crashes?
The bank runs were not caused by the crash; they were both caused by the same phenomena.  In the run-up to the crash banks around the United States sent requests to city banks and the reserve banks for gold.  In part due to the poor information collection system at the time, there was no way for the Fed to distinguish between a heating up economy and a true shortage of gold.  

The Fed interpreted the requests for gold as a heating up economy and so contracted the money supply by 3%.  This exacerbated the shortage of gold around the United States.  It quickly propagated through the world financial system because of the shared gold standard.  The market crashed because banks were issuing margin loans at up to 90% of value.  The older system in use, before the Fed, was to create a banking suspension where banks would collect gold, but not disburse gold.  The Fed made that functionally impossible forcing runs.

The thing to remember about stocks is that they are purchased with your spare money.  Given a choice of eating food or buying shares in Apple, people will choose food.  When the money supply contracted, funds didn't exist for businesses to continue operations, to pay employees, so people sold stocks.  The sales forced margin calls, which triggered more sales, which triggered more margin calls and so forth.

The NY Fed did provide liquidity at the beginning of the crisis, but the Board of Governors were still convinced that the contraction was necessary and forced the end of the NY Fed's liquidity provision.

The stock market was the thermometer, not the cause.  The system had become sick.

To be fair to the Fed, it was itself part of a much larger set of global events without an information base to act on.  It was also true for the Bank of England or the main European banks.  The gold standard propagated economic issues in ways that were not understood at the time. 

When World War II ended the nations of the world created a variety of treaty organizations such as the United Nations and the OECD to prevent such a sequence of events happening again.  The problem is that restricting those events from happening means that some large business owners miss out on opportunities and so there is pressure to dissolve or weaken the treaty organizations.  In addition, the major nations sometimes endanger the system.  You can see this very directly through the actions of the Trump administration.

The postwar era institutions really do need reform as the threats have changed, but the political will to protect the world doesn't play well in local political processes.