- Shut the door, will you?
- Shut the door, won't you?
I can't tell which way is the correct way. Both of these sound correct to me.
Both #1 and #2 are okay and mean approximately the same. "Shut the door, won't you?" is slightly more polite, because the tag at the end suggests that you might not be willing to comply, and that is a conventional way in English of suggesting that if you don't comply with a request, that is perfectly acceptable to the person making this request. It's not a demand. A similar way of offering in advance an excuse for not complying is to add on "if you don't mind", "if you wouldn't mind", "if it's not too much trouble".