Since India announced a strict nationwide lockdown in March, the financial system has been buttressed by emergency liquidity injections, credit guarantees and a temporary moratorium on loan repayments that runs through August.
With the virus still raging — cases in India exceeded 1m last week — Mr Kumar said particularly vulnerable industries like aviation, hotels and jewellery are candidates for further support.
"We will take all necessary steps to ensure that we're able to help out whoever needs it," Mr Kumar said. "There are many people who have good intentions but may not be having enough cash at their disposal."
Sunil Mehta, chief executive of the Indian Banks Association, which Mr Kumar chairs, said he has submitted a proposal to the government to extend tax relief and other emergency measures to airlines and others lest they collapse and contaminate lenders' loan books.
"Every impact on the economy is transferred to the balance sheets of the banks," Mr Mehta said.
Even before the pandemic, the financial system had been rocked by a series of near-misses. In March, Yes Bank, a previously fast-growing private lender, was the subject of a central bank-led rescue by SBI and a consortium of investors.
While the system-wide ratio of non-performing loans had eased from a 2018 peak of 11 per cent, credit rating agency S&P expects it to rise back as high as 14 per cent in the financial year ended next March.
Fitch estimates Indian banks will need anywhere from $15bn to $58bn next year to withstand the shock to their loan books.
SBI said last week it planned to raise Rs200bn ($2.7bn) in equity capital, joining a range of other private and public lenders like ICICI, Axis and Punjab National Bank in seeking to bolster their balance sheets. But weaker public-sector banks may require government support.
Mr Kumar said the "real test" would come later this year.
"We're in wait and watch mode," he said. "Depending on how fast the recovery happens, the capital buffer and the speed of resolution of stressed assets . . . can enable the financial system to deal with this current situation."