If you do destroy it and it doesn't trigger at all it's two gold and a card that your opponent just wasted. You're hugely ahead on economy and board position now as whatever you attacked with is on full health. Probably from this position you win the game. That's how far ahead you are if you make your opponent waste two gold on turn 3 or 4.
Ok, now assume you don't destroy it and it triggers but doesn't die. Ok, now you're behind by one gold and unless you kill it this turn you'll be behind by four gold. If it lives again you'll be behind by seven gold. You can't fall behind to the tune of seven gold. That means they're ahead of you by a whole max band hero.
Killing a mine without it triggering is huge value. You just cost your opponent three gold (actually cost them five gold since they spent two to get the mine). That's the main reason mine is such a weak card. The cost of using it if it doesn't work is crazy high.
If you break through the patrol zone you should always kill Rickety Mine if you can. You can always do another two damage to base. Probably you can't survive falling behind by 1.5g per turn that mine is in play.