If you are using the Windows operating system, try to use the Firefox Browser because it is more secure. If the computer doesn't already have Firefox on it, you can download Portable Firefox to the desktop and run it without needing admin rights to install it.
Turn Off Auto-Complete
Make sure that the browser is set so that it does not remember your passwords. You can set that option in the settings from one of the menus. The exact location of those settings is different for each browser.
Type Your Banking Site's Address Directly
Don't visit your banking web site from links. Always type the web address into the browser's address bar yourself. This will help reduce the chance of phishing scams where people are redirected to sophisticated fake bankings web sites that can look exactly like the real thing.
When your connection is encrypted the http:// in the web page address should turn to https://. In Firefox, the address bar will also turn yellow. Below is an example of an encrypted connection to a banking site on Firefox (running on Ubuntu Linux).
Note that the web address starts with https:// and that the address bar is yellow. I've highlighted the https:// part:
If you look at the status bar on the bottom of Firefox, you should also see a lock which also shows that the connection is secure:
Other browsers should have similar features. Look for the https:// and the icon of a lock in the status bar at the bottom of the browser. I am using Firefox as an example because it can be found on most Linux distros, and can be run on most Windows machines by downloading Portable Firefox and running it from the desktop.
Troubleshooting Encrypted Connections
If you go to your banking site and it doesn't show you the encrypted https:// connection, type a few random characters into the username and password fields and submit them. The page will reload, telling you that the login was incorrect, but the page will now show you the encrypted connection.
This is a new feature of some web sites and can be ignored if the page shows an encrypted connection after you submit the random login information.
Closing Your Banking Session
Always log out of your account when you are done, and also be sure to close the browser to remove additional information that may be stored in the browser.