Hey Jim,
This works this way: After signing in on the computer, if you sync with any device (like your son's iPod Touch), the credentials will be copied to the iPod and therefore anyone will be able to access the App Store using your account on the iPod.
So, you should do the following:
➡ If you want to sign out on the computer, simply find you e-mail address on the top right bottom of the window, click it and then click "Sign Out".
➡ On the 3.0 version, there will be an option where you can also sign out manually. However, at this time the only thing you can do is block access to the iTunes and App Store apps on the iPod. To do so, just go to Settings.app -> General -> Restrictions.
So, all this means that turning the device off after each download won't disconnect it from the Itunes store.
Hope this helps!