More seriously, you don't need to do extra due diligence. Just buy insurance for your funds above $250K. Or use standard treasury methods like using institutional insured liquid deposits or sweep accounts.
Liability insurance is required for automobile drivers, because of the externalities. Perhaps bank run insurance should be required for depositors, because of the externalities. That'd force depositors to internalize the externality of banking with a fragile bank.
More seriously, you don't need to do extra due diligence. Just buy insurance for your funds above $250K. Or use standard treasury methods like using institutional insured liquid deposits or sweep accounts.