I have an external USB drive that I use for Time Machine backups. I've noticed that when I right-click on an item the Open With menu lists not only apps in my Applications folder, but also apps in the Applications folder on the Time Machine backup. Rebuilding the Launch Service Database will help clear out the list, but then it just grows back. I attempted to exclude the external drive from Spotlight indexing, but the external drive doesn't appear in the list of drives to exclude. Below is a screenshot of my Open With folder.
Best Answer
You should be able to exclude Spotlight from adding the apps on the Time machine drive to the "Open With..." menu by opening System Preferences -> Spotlight -> Privacy. Then drag the icon of the Time Machine drive into the "Prevent spotlight..." window. This adds that drive to list of drives that Spotlight should ignore for searching operations.
Note that you can't prevent Spotlight from indexing that dive as Time Machine depends on Spotlight to keep a catalog of files, but if you do this and then rebuild the Spotlight index it should prevent Spotlight from adding apps on that drive to your "Open With..." menu.