When I find a cavern I plug the hole ASAP, typically by building an Up stair (as I normally find them by digging staircases). Once that's taken care of you can plan your cavern breach in peace and quiet (providing you don't get visits topside).
My careful/whimpy approach (there are others) is:
Dig a tunnel up to, but not through the wall where you want your proper cavern entrance to be. I then build a raising drawbridge (2 tile long to visually see whether it's up or down) by the wall, 3 cage traps inside that, an empty spot for a future upgrade with a repeating menacing spike trap (for FBs), another empty spot that could hold a cage trap, a wooden door (bait for FBs and building destroying critters you actually want to trap in the cage traps, as well as something you can lock as soon as the last dorf is inside when evacuating the cavern, both because it will block/delay anything following them, and to stop the cancellation spam resulting from the civilian alert). Inside of the door I place a second raising drawbridge (again two tiles long). The bridges are hooked up to levers, and the hookup tested (it's no fun to find you've accidentally built retracting bridges instead when you really need them raised).
Then I wait patiently, observing the cavern wildlife in the parts I can see. After a long wait (typically) I find that the cavern probably has a benign critter group that entered recently (and thus won't be replaced immediately with something nasty), and then I send a miner to dig out the last tile to breach the cavern, as well as a militia squad (typically consisting of a single dorf who's been assigned the spare axe, but with no training) stationed by the airlock. I also set up a civilian alert to exclude the cavern from civilian access so they won't all rush out and start to collect old desiccated skeletons). Once the miner has breached the cavern I station the militia at new positions in the cavern to explore it fully, retreating at the sight of anything dangerous for a new attempt later, but normally able to explore the cavern fully at the first attempt. As soon as the cavern is explored, the militia is restationed inside of the airlock, and the outer airlock drawbridge is closed. The door is locked behind the militia once inside (to cut the spam), and once the outer airlock bridge is closed the militia orders are removed, so the dorf can go back to the normal duties, and the door is unlocked. The civilian alert can be lifted as soon as the door is locked.