Maintaining a secure computer takes a bit of work - you need to have a firewall, run virus scan every so often, download new updates to your operating system, internet browser, or other software... Can I trust the owner of the computer has taken care of that?
Can I trust that the computer I am using is clean? Many people download and install various programs (people especially like free software) on their computers all the time. That software might have had some spyware bundled with it. Lots of people also like to open cute little email attachments not knowing something else is also attached. It's so easy to have some spyware/virus put on your machine. If you click on a malicious pop-up ad or surf to some bad site, something might get installed without you knowing.
Can I trust that the computer administrators have used strong enough passwords and maintained their computers sufficiently to protect their accounts from being hacked? Can I trust that some smart kid didn't install some spyware on the library or hotel computer without anyone knowing?
If there is some spyware installed on the computer, information about me - where I surf, what I type (including passwords, account numbers...) - could be recorded and sent to Mr. Stranger.
Hopefully one day, they (computer makers/software developers) make it easier to keep your computer secure.
When using not-my-own computer, I typically just surf and avoid tasks that involve me entering in sensitive information (like logging into email, online banking...) If I'm traveling and I don't have my own computer or Blackberry, and I really have to check email, I'll usually change my password when I get home, just to be safe.
