Based on past behavior of the Department of Treasury and the Federal Reserve, particularly during and right after the 2007 financial crisis and resulting Great Recession, the Department of Treasury would most likely try to find a larger financial institution to absorb the smaller, less well capitalized bank. This happened numerous times during the financial crisis: Bank of America swallowed Merrill Lynch; Wells Fargo swallowed Wachovia, et cetera.
Keep in mind that since FDR's New Deal and the creation of the FDIC (Federal Depositor's Insurance Corporation), the federal government has required all banks to contribute monthly fees to an insurance pool that makes depositors whole in the case of a bank failure. Currently, as of 2016, in the case of such a bank failure, each depositor at a given bank is insured up to $250,000.
Since the passage of the financial form act known Dodd-Frank, however, banks are no longer supposed to be bailed out if they fail. In fact, such an action is supposedly illegal. So, if the bank is a small one, the likely action by Treasury and the Fed would be to arrange a fire sale of the bank assets by another institution, as mentioned above.
If the bank in question is larger, however, and its failure is considered a "systemic risk to the economy," the Fed might well be forced to open its special lending window to recapitalize the bank, and to protect the bank's "counter-parties," meaning the other institutions it owes money to or insures.
This action would constitute a bail out, and because of the consolidation of the banking industry during and after the financial crisis, the effects of letting the bank go under might be too catastrophic for the Treasury, and whoever occupies the White House, to contemplate. In that case, the president could convince Congress to give the bank in peril an exemption so that the Fed and Treasury could begin infusing the bank with funds.
Even so, if such a bailout happened again, political pressure would most likely force the terms of such a bailout to be unfavorable to the bank in peril and its executives, forcing out top management and giving the government (and taxpayers) ownership of the bank in return for saving it.
Whether that kind of action constitutes "helping the bank," however, is a matter of one's perspective and political philosophy.