You have to build their trust in the system, at all ends of the transaction. Let them know about how many large companies, Bestbuy, Barnes and Noble, Walmart, Circuit City, etc. they all have websites, and they offer their products online. They wouldn't allow their good reputations to be tarnished by an unsafe web right?
You need to convince them that the computer is safe. Show them that you've taken time to ensure that it is virus-free, that there are protections in place, that nobody can secretly steal their information on the home computer, etc. Inform them of safe practices they can follow to keep their computer fraud-free. Even show them that you can have the private information deleted so there is no residual trace of the transactions.
After this, you move onto the technology itself. Tell them about the encryption standards that all browsers have. Back in the old days, you had to agree to some weird license agreement to even use 128-bit encryption, but now all browsers have it. The encryption is so strong that, even if someone were to intercept the transaction, it would take some odd billion years for every computer on earth to break the encryption. It's why you can buy things over open wireless connections and not fear your information would be stolen. Show them the lock icon in the browser window that shows that the session is encrypted, and that all major sites that handle credit cards will be encrypted. Tell them that they should look for such an icon when they visit a secured webpage. Even go through the trouble of showing them the signing authority (click on the lock), and how they must be certified before they can handle such transactions. Even show them how an encrypted session with a non-signed authority will appear, and that your browser flags such site for your protection.
Finally, ask them whether they monitor their credit card statements for suspicious activity, and how they react to unauthorized transactions from any source, online or not. One of the reason people use credit cards, aside from convenience, is because in the event of a fraudulent transaction, a product dispute, or even just a grievance, that customers can dispute and cancel charges with rarely more than a simple phone call. Inform them that this policy is taken to a fault with many sellers being denied their payments by unscrupulous customers. The scale tips heavily in favor of customers.
For a first run, ask them to try it out with a big company, like Circuit City. Tell them you want to buy a DVD-RW drive or something (so you can extract the diode right?). Rather than pay for it at the store, have them buy it online and you pick it up from the local store. That way there is a connection between local store and online transaction. It'll also make them feel safer because they can go and yell at whomever is at the counter at Circuit City if something goes wrong. Show them just how convenient it is that you can search for whatever they offer without having to mope around the store too. Furthermore, visit a site like Amazon and show them all the products that you can't get locally, and the great prices you receive there.
Another thing you should do: tell them to read up on it themselves and not just take your word for it. Have them talk it over with their bank official, or visit Visa's website to see what kind of protection they receive on their credit cards. The best person to convince them is themselves.