"Put" means to place an object somewhere.
"Keep" means to store an object somewhere on a permanent or long term basis.
I put my bag on the desk so the inspector could check it.
I keep my bag in the closet when I am at home.
The first is temporary. The second is more or less permanent. You can use 'put" for long term placements, but you cannot use "keep" for short term placements (unless the position of the bag is not changing, in which case it is a different sense of "keep"). "Keep" also implies possession, so you would not use "keep" of someone else's thing even if you were placing it somewhere long term.
If you want to place the box temporarily somewhere, or for an unknown period of time, or it is someone else's box, you say:
Where should I put the box?
If you want to store it permanently somewhere and it is yours (or you are looking after it) then you say:
Where should I keep the box?