How quickly do you want to be "ready to go" each time you step up the the Karl Fisher and want to run a test.  If this is important than the other issues are probably not as important.   

Keeping atmospheric moisture out of the cell by either keeping the instrument running or by waiting to turn the instrument on is going to require possibly about the same amount of energy and reagent to bring the vessel to ready mode.  I guess you could experiment with ways of storing the vessel while it is still charged with reagents - in dry places, etc...but I don't know if that will really work.  I guess alternatively you could use an extra reagent bottle to store the in-use reagents and see if that works - problem with this though is you end up mixing up good and old/bad reagents and may waste reagents in the end...

As for the instrument there really is not a problem by keeping it on as the stirrer motor is probably the only mechanical part that stays running....kind of like a fan on a PC.  I don't really think this is a problem though.