I almost always use a "Virtual Credit Card" from CitiBank over the internet, or by FAX and often by telephone. It's a one-shot use, only good for one month max credit card number. But it doesn't work where you have to show your credit card matches that number, (like airline reservation e-tickets) because it won't.
To the question, I recently had an occasion to send some dive equipment to a purchaser in New Zealand. He paid by PayPal, but I was curious why so many would not participate in overseas transactions (besides the little be more effort to prepare the customs forms) so I went to a friend how manages the shipping depart for a major Money Museum. He said that, unless they are repeat customers, he always requires a valid photo ID (Passport, Gov't ID, Drivers License, etc.) showing the same name as on the credit card, and the same address as he's shipping to. When I asked why, he replied that the Credit Card abuse rules are for USA only (although there are other countries that also have rules). It is known that someone will steal a credit card number from someone, and order internationally with that number and name, and have it shipped elsewhere. There is no universal (international) protection against that, hence the additional validation. He doesn't ask for the Credit Card pictures, but the name on the Credit Card account and the name on the ID must match, as must the addresses. I suspect it's not 100% foolproof, but it certainly lowers risk significantly.
I would have no problem Faxing most anything, for the prospective of "theft enroute". But I would never email my credit card info. Well, I have, but I do it in several emails, one with my name, another with the number, and a third with the expiration date, and, if required, a fourth with the CVC code. Chances of an internet sniffer picking up all the emails, none of which have redundant info, and piecing it all together is very low. Chance of intercepting a FAX is very low.
As far as misuse of the Credit Card number once it's at the destination, it is no more likely than if they had a just the number. I think we should minimize the risk, but the only way to completely remove it is to do away with the Credit Card (and we are only eliminating credit card risk then).
As an aside, on a cruise I took several years ago, the cruise line suggested that I make a copy of all my credit cards, my driver's license, and my passport so that if I lost my wallet (or it was stolen) it would be easier to replace. I scanned all the documents, printed them out, and then said to myself "this is exactly what I would need if I were to steal someone's identity, or use their credit cards". I immediately shredded them.