Research by analyst Gartner found that ensuring they were able to carry out transactions over the internet without fear of exposing their personal details was the biggest concern among UK and US users.
Other factors found to be important include the ability to set up alerts showing debit and credit card balances, as well as a single sign-on facility.
Research director at Gartner Stessa Cohen said the findings highlighted differences between the demands of banks and their users.
"Customers want to personalize their experience ... so they can better manage their money, pay bills and transfer funds. Banks want to sell customers more products," she explained.
Ms Cohen warned that banks that do not offer their clients customization options now may find that they begin to see a decrease in the number of customers using their online and mobile banking services.
In related news, Visa recently announced it is trialing a new system whereby customers are notified of transactions by text message.