Online banking saves the bank money -- why would I pay for that? They should pay ME to use it!
Answers

I've read many times that online billpay is costly to banks (averages 40c per payment + the costs of subscribing to the service, nearly all of which is outsourced), yet they've discovered that those who adapt to it are likely to carry high deposit & credit balances.
In 2000, I specificity chose BankOne as my first bank as it offered free online banking and free online billpay, unheard of at the time. Of course, this was during the era of soaring bank fees and it didn't last long and I immediately left BankOne. (I cringe when I read that free checking has been a mainstay since the 1990s when in fact the late 1990s/early 2000s had among the most & highest fees of all time.)
But to answer the question, if banks started charging fees for online banking, I'd have to weigh the convenience & cost vs. the alternatives (not all bills can be paid online, and some creditors charge recovery fees).




Over my dead body. I'll rather start mailing checks again (never did much anyway back in the days) and support the USPS and make it more expensive for banks by them having to handle more checks;-p