Reading this reminded me that I hadn't connected my Garmin to the Mac since I upgraded to Mavericks, so off I went and connected my Edge 205 and went to Garmin Connect. Safari came up with a warning asking me confirm that I was happy to use the communicator plug-in, I did and it hen worked normally.
So, I can't help you with your problem but I can confirm that Garmin Connect does work with Mavericks. My Mac is a 2011 iMac.