No, it should be perfectly fine.
If you're ever typing secure information, make sure that you see HTTPS (note the S) at the top-left of the address bar, and also make sure that you see a padlock icon too.
The padlock icon will be either at the top near the address bar, or in the status bar at the bottom (depending if you use IE or firefox or safari, etc.).
Under HTTPS, your details are secure even across an insecure wireless network in your library. People will know that you've been to your bank's website (though people can always look here, even in a cybercafe, it's trivial if they want to), but your details will remain secret.
If you want to send secure email, then I would suggest you look into setting up GPG with your email client on your laptop. This is easy to do and bulletproof security-wise, but also requires the recipient to use a similar system.