The most important quality would be reliability. Since there is real values involved the tolerance for mistakes is really low. It has to function properly no matter what and since there is private information involved it has to be secure. Other than that, a large part of the services should be given on the internet, or at least in combination of internet, phone and mail. Customer should be able to reach any service he needs without going to the bank in person. Then there is the ease of use and accessibility of course as same as any other internet system.