The answer is easy. Look at the number of security breaches every week where a hacker gets access to credit card or personal data from on-line sources.
While you may not have to pay for fraudulent charges (IF you can prove the charges are not yours) the identity theft problem is enormous. Some identity theft victims have had to battle for up to 7 years to get all the fraudulent activity off their records and clean up their credit reports. Some victims have stated they have been unable to ever correct all the errors.
So, we don't take credit cards over the internet at this time. Maybe one day, when it can be guaranteed safe.