What is the objective methodology to tell the difference between a "bad pun", and a "good pun"?

There is no objective method, but there are good puns as opposed to bad puns. A bad pun is one that is forced, while a good one is a witticism where the listener does not feel punished.

With due respect to @zenvelo, it’s not that black and white, not that defineable. My father was a great/terrible punster, and his family and friends routinely thought his “bad” puns” were as funny or funnier than his “good” ones.

Having followed in Dad’s punsteps (get it?), I’ve told and heard a LOT of puns over the years. I’ve been giggled at, groaned at, ignored entirely, and often—and as recently as two nights ago—enthusiastically congratulated on the quality of my pun.

A pun isn’t good or bad simply in and unto itself—its humor, or lack, derives from the delivery of the person telling it, the situation and environment, the sophistication or non-sophistication of the wordplay, and not least, the kind of sense of humor the listener has.

Other jellies may be able to objectively explain the difference between a good pun and a bad pun, but as for me, a punster all my life… I know the difference when I say one and when I hear one, and, like my dad, some of my worst ones (and there have been a dazzling number) have been great.

The best puns are the ones greeted with loud (clearly envious) groans.

Puns are strictly subjective.

A good pun is one that I make that makes the listener groan.

A bad pun is anything that anyone else says.

If you want to punch the person in the face it was a good pun.

A good pun is its own reword.