There aren't a ton of us who use public computers; you know, the ones you find at libraries and ... well, libraries. But to some degree even the computer you use at work is a semi-public computer, and odds are it's got a ton of your personal information on it.
Admit it - you've bought something online at work. Or at the very least logged in to your bank account or made a credit card payment. We're at work 40 hours a week or more; sometimes, you've got to squeeze in one or two virtual errands from the office. We get it.
While the desktop computer you use at work is obviously more secure than the one at your local library, you should always make sure that your password to log on is tough to break and that you NEVER save any sensitive passwords in your browser.
But for highly public computers like the ones at the library, it's recommended that you keep any and all private documents off of these computers. Putting your credit card number into public domains puts you at even further risk for ID fraud; it's best to just avoid this scenario entirely. And on a similar note, make sure you're checking your credit card account regularly for fraudulent charges. The quicker you identify ID fraud, the quicker it can get resolved without implications to your credit report or personal finances.
Keep your credit card numbers as private as you can by isolating them to your personal computer and smart phone only.