I've banked online for years, though I did resist it at first because I didn't think it would be safe. I do it on my mobile and my computer. I think that, provided you don't save your password anywhere, it is pretty safe. However if you save your password on Google and lose your phone you're just asking for trouble.
I very rarely use cash for anything now and they are threatening to close the only local cashpoint so I really don't see how anyone can avoid using internet banking in future - though I do understand how anyone who doesn't use computers must be very nervous about doing it.
My mother (sadly no longer alive) had terrible trouble remembering her PIN in her 80's and always had to go inside the branch as her card was swallowed every time. The cashiers were lovely and knew her by sight so there was no problem.