For your consideration ...
I never do any on line banking or anything else on line since a FBI Agent I know had told me anything Internet based can be broken in to at any time. It is just a matter of how hard the person tries.
That would mean that it does not matter that you don't bank online. The bank itself conducts business online thereby putting you, and everyone else that doesn't do online banking, at risk as well.
As a thinking exercise, think about this ..... If all banks computer memory, hard drives, backups, etc, were to bite the dust and end up in the bit bucket, taking every account with it, would there be, are there, any physical records, ledgers, of anyone's accounts held at the bank, as was in the past? Hummm.
That's the funny thing with fiat currencies. "Gold is money. Everything else is credit." ~ JP Morgan 1912.