This is why we adopted “tiering”, or allowances up to which banks can deposit excess liquidity with the ECB at 0% interest. A monetary policy problem would arise 3/6  BIS central bankers' speeches  if the banks became reluctant to grant loans due to excessively weak earnings. But we haven’t observed that yet.
For all these reasons, my strong preference would be to use regulation and supervision to strengthen the financial system and lean against developing problems. Given our current state of knowledge, monetary policy would be used only if imbalances were building and regulatory policies were either unavailable or had been shown to be ineffective.