I've heard all kinds of 'I don't trust it' stories over the years.
In the 90s my credit union was the first bank I heard of to go online. I jumped on that immediately, but most everyone was nervous about it. I thought.. "why would a respected financial institution do this if it was fraught with errors?" Answer: they wouldn't.
Then came Amazon, and all the fears about having your credit card numbers "out there" for all to see. Again, no problem.
Just a few months ago I talked to a man who mentioned he had to visit the bank. I made a comment about "who goes into banks anymore?". Well, he does, because the online stuff scares him.
In 25 years of online banking, online shopping, depositing checks over the phone, I've had exactly ONE error, where the bank credited a $600 check as $60. I guess I can live with that.