When I close a Netflix tab on my browser (Chrome), the Silverlight plugin process stays running. I can quit it manually using Activity Monitor (I'm on OS X), but it would be nice if it quit on its own, since my computer uses its dedicated graphics card while Silverlight is running, which wastes battery. Is it possible to have Silverlight quit as soon as I close a tab that uses it?
This sounds like a bug in either Chrome or Silverlight. Usually, the plugin does close/gets closed once you close the Netflix tab. The fact that this doesn't happen indicates a fail in the browser or the plugin itself.
To pinpoint the source of the problem you could try a different browser and see if this fixes the problem. If this is the case then it's a bug in Chrome. If not, then it's Silverlight.
The obvious solution would be to check for updates for both the browser and Silverlight. You could also try to downgrade either one. Knowing when the problem occured first could be the key here. Did you upgrade either components before it happened, or did you install any addons?
Another tip would be to use Chrome's incognito mode. It disables all add-ons. If the problem still exists then you can rule those out.