I too am plagued with this problem.First of all on my Android 2.3.6 phone, even after a fresh restart there is an Airdroid service running taking 4mb of RAM. This is an older phone and I need every MB of RAM available. I've used an autostart manager program to disable it, but it still starts on startup. Also I when I try to stop it under Settings->Applications, nothing happens.
On my Nexus 7 with 4.2.2, after running Airdroid and exiting, the icon remains in my notification bar (and I'm sure still in memory too, haven't bothered to check since I'm not strapped for RAM on this device). This is VERY annoying. When I exit a program, I want to EXIT. I have to reboot my Nexus 7 to get rid of it!
Other than that, it's an amazing program, simply incredible.