AbeBooks is a very secure place to shop. If you are paying with a credit card, their external paymnet service provider processes your payment. Their external payment service provider stores your credit card details on their secure server, and they are never shared.
They employ SSL (Secure Socket Layer) technology to keep your credit card information safe, and use encryption to protect transmitted data. They will never pass your credit card number through email.
To ensure that all of your personal information is kept safe, it is important for you to sign off once you are finished using the website, and close down the browser window if you are on a public or shared computer.