Web browsers help you remember user names and passwords for banking accounts by keeping them in memory. Even better, your browser will fill in passwords automatically for you.
The problem arises when you donate or recycle you old computers. The trusted charity, electronic recycling company and anyone else with access to your computer also gets full access to user and passwords.
If I have access to your PC, I bet I could obtain access to your banking accounts withing minutes.
See the following USA Today article on how to view how to do it for each popular browser. Try it for yourself and you will never allow an old computer out of your site.