It's important to note that your creditcard is really no less secure over the phone than it is online. And often times, it's the person(s) at the other end of the transaction that you have to worry about more than someone who may or may not be close enough, and paying enough attention, to actually hear what you're saying. The only potential advantage to doing it online, and it's only a slight advantage, is you can do it at a time when there's no one else who could possibly walk by and hear, or see, roughly what you're doing. And I say slight advantage for two reasons. First, as stated that assumes the people nearby both are close enough and care enough to actually hear and either acknowledge or remember what's being said, or what they've accidentally managed to catch a fleeting glance at on your monitor on the way by. And second, while it eliminates potential problems that may or may not be significant threats from the point of origin, it does nothing to prevent your creditcard info from being misused or otherwise at the other end, as well as opening you up to the possibility of it being intersepted by a third party--the so-called man-in-the-middle attack.
Now, I'm not saying all that to make you more paranoid than you may or may not already be, but rather to point out the concept of absolute security/privacy is really a myth. If we all stopped doing things or avoided doing things a certain way because some piece of information may or may not be overheard by a person standing 10 feet away, or even because someone happened to be writing it down as you were giving it to them so that it could be used later, we'd never get anything done. That includes in person if you happen to not be able to see what's going on--you could hand someone your creditcard and while you're receipt's being printed off, they could be writing down the info off of it. But, just because it could be happening, doesn't mean your chances are anything beyond extremely slim.

So, in shorter terms... stop worrying and pick up the bloody phone. You're not losing much.