HTTPS encrypts data as it is sent across the network. So, using HTTPS while sending your password to an Internet banking site would prevent a hacker from seeing your password as it crossed the network. However, the browser is responsible for managing how it stores data in memory. If you're using Internet Explorer, it may solve the problem if you disable IE's ability to remember passwords--it will always prompt you before it remembers any password.
This was first published in October 2001