There are advantages to upgrading to a smartphone. It allows you to download the bank's mobile banking app, which eliminates the need to type in the bank's URL. It also shortens the steps to get to your bank's website.

Note that some banks have a special URL designed just for a mobile phone's small screen. Some banks, such as Citibank, will automatically detect mobile devices and take you to a different home page.

For security reasons, make sure that you download the app only through your bank's website or from your phone carrier. "The easiest way to find your bank's app is to go to the app store of whatever phone you own and search your bank's name," Bissell says.

A downloadable app gives you a better interface for accessing your bank account remotely. "Sometimes the app is easier to use than the regular website," Bissell says.