The falls come after authorities moved to protect customer deposits when the US-based Silicon Valley Bank (SVB) and Signature Bank collapsed.
Joe Biden promised to do “whatever is needed” to protect the banking system.
But investors fear other lenders may still be hit by the fallout.
On Tuesday, Japan’s Topix Banks share index fell by more than 7%, putting it on course for its worst day in more than three years.
Shares of Mitsubishi UFJ Financial Group, the country’s largest lender by assets, were down by 8.1% in mid-day Asian trading.
On Monday, Spain’s Santander and Germany’s Commerzbank saw their share prices dive by more than 10% at one point.
The volatility has led to speculation that America’s Federal Reserve will now pause its plans to keep raising interest rates, designed to tame inflation.
Mr Biden said that people and businesses that had deposited money with Silicon Valley Bank would be able to access all their cash from Monday, after the government stepped in to protect their deposits in full.
Many business customers had faced the prospect of not being able to pay staff and suppliers after their funds were frozen.
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BBC North America Technology correspondent James Clayton spoke to people queuing up all day outside the SVB branch in Menlo Park, California, to access their funds.
As the bank was no longer offering wire transfers, they were taking out their money in cashier cheques.
