You can't. It's a security feature of the FlashPlayer to stop rogue developers from being able to access your microphone or web cam without the user knowing. Could lead to all sorts of dodgy apps out there.
Maybe you can have your client copy/install the flash to their local drive and circumvent the restriction this way?
I've never encountered this behaviour so I can only guess as to the reasons. My first hunch is that Flash player considers it to be a different swf each time. That would mean it's not really "forgetting" what you set, but rather not thinking you've approved that specific swf. Maybe looking into what it uses to identify the and link the approval to it could lead to some progress?
Bah, how stupid. The problem was indeed our fault -- a fellow developer had added a call to Security.showSettings() during one module's initialization function, apparently to fix another bug having to do with getting the microphone delay time. Doing this will show the dialog regardless of what the user had selected for the "remember" checkbox.