Searched with no luck. I no longer have the 610 ((upgraded to the fenix :D)). Very annoying that basecamp keeps on trying to sync with the 610. Uninstalled the ant agent and basecamp. Reinstalled basecamp and the 610 still shows. I do not see any options to remove this. Has anyone had any success with this?
I delete all contents under the directory located at: your user name, Library ((it's a hidden directory)), Application Support, Garmin, Devices. I only had 1 device, if you have multiple devices you might want to procede with caution. But the annoying sync went away for me and everything still works fine.
I do not have my Forerunner 405 any more, but BaseCamp (MAC version) continues to display the device. You suggest to remove the device in ANT Agent. So I installed ANT Agent again but to start it I need the stick that I do not have any more. So impossible to go farther. What do you suggest in this case to remove the old device?
Open Finder and navigate to the folder /Users/{yourname}/Library/Application Support/Garmin/Devices. If you can't see the user Library, hold down the Option key on your keyboard, open the Go menu of Finder and select Library.
In this folder you will find one or more folders named with a number. This/these folder(s) contain(s) a local copy of the contents of an ANT-supported device, and this is what BaseCamp uses. When there's only one folder you can safely delete it; otherwise navigate into that folder and open the file GarminDevice.XML in Text Editor. Somewhere near the top of that file you will find to which device it belongs.
Edit: I didn't read the whole topic before answering and obviously you didn't either, because the same answer was already given...