There is no real off switch for the hopper or joey as they remain almost fully powered 24/7. The off switch just turns on the stand by screen and frees up any live tuners in use.
The only time you have to keep the Hopper on is if you paused a live tuner and wanted to pick it up on the Joey in another room. In that instance if you power off the Hopper you lose your pause point in the program. Other then that, state doesnt matter.