In your browser settings make sure you have the option 'don't save secure pages to disk' - this prevents IE from saving SSL files (browser pages you visit, like the banks) in your 'temporary internet files' cache.
Make sure your browser is 128 bit (choose Help About to verify).
Make sure when you visit such a page, that the little yellow LOCK is LOCKED (bottom right of browser window).
When the yellow LOCK IS LOCKED in your browser, any form communications/passwords, etc. is sent to the other server ENCRYPTED. Even if someone did get a copy of the file from your HD, they'd have a heck of a time UN-ENCRYPTING it. It would take years with a single PC.
I would never give Credit Card Info on the net WITHOUT the SSL LOCK.
SSL is 'Secure Sockets Layer' - it encrypts ALL transactions sent between the client (YOU) and the server (THE BANK, STORE, etc).
Really though, I'd be more concerned about someone hacking the server site, and obtaining their list of credit card numbers (including mine) than I would someone obtaining a file from my hard drive. Why go after ONLY one credit card number when for the same amount of trouble you can get a stores/vendors entire list of customers and their credit cards by stealing their database.
This HAS happened to AOL and smaller e-commerce sites in the past. I wouldn't worry about banks though.
That's about as secure as you can get -- it IS safe, just make sure SSL is being used (the LOCK is LOCKED).
Hope that addresses yourconcerns.