It's a good question and the answer is that the situation is unstable and dangerous. But I think you are approaching it backwards:
With almost every country in the world, the US has first strike capability - the US could wipe out the country in hours or less, and only a few countries have a second strike capability to deter the US.
That had long been true with NK, a very belligerent enemy. But in the last couple of decades NK added a small nuclear arsenal. It's not enough for a MAD relationship with the US, but they could threaten great harm to US allies South Korea and Japan - imagine nuking Seoul and Tokyo - and possibly land one on US territory. It wouldn't destroy the US, but losing San Francisco is a serious deterrent.
Did the addition of NK's nuclear arsenal stabilize the situation by creating more deterrent, or destabilize it by emboldening NK? It's complex:
One factor is that the US has sworn off use of nuclear weapons in conventional conflicts, even ones they are losing, and have strictly adhered to that policy, not even using small tactical nukes. The US has an even stronger motive - it establishes a global taboo against nuclear weapon use that nobody has violated yet. NK is very aware of it because US generals recommended using nuclear weapons in the Korean War and the president declined - that may seem like too close a call for NK, and don't assume that NK understands the US nearly as well as you do (if you are American); miscalculations like that are common on all sides in international relations.
So now that NK has nuclear weapons, does that make a conventional conflict into a nuclear one, destabillizing the situation? What if the US believes they need to use nuclear weapons to prevent NK from using them on Seoul or San Franciso?
On the other hand, NK's nukes may prevent a conventional war. NK saw what happened to Iraq - everyone did, and many realized that actual nuclear weapons were their only defense if the US was going to ignore international law and sovereignty and engage in 'regime change'.