More nuclear powers mean less stability.
There used to be two. Then five or possibly six. But France and the UK were closely allied with the US, so adding them to the mix didn't complicate things very much. The US and the USSR faced each other with vast intercontinental arsenals, the Brits tried to menace Moscow in case Washington got wobbly.
These days, Chinese missiles menace the US and India, and possibly Russia, India menaces China and Pakistan, and so on. So if the US were to install a missile defense against North Korea, China might be prompted to upgrade their arsenal, which would worry India. (There was a suggested edit that China and Pakistan have promised No First Use. That's not the point of this part of my answer. I'm talking capabilities and and the direction these warheads could potentially be targeted, not intentions.)
If North Korea gets an operational nuclear force, both South Korea and Japan may be prompted to go nuclear themselves. If Japan becomes a nuclear power, there will be political earthquakes.
Of course it would be completely unfair to tell North Korea "sorry, the club is full," but this is not the main reason to worry.
North Korea, in particular, is seen as a Rogue Nation.
The North Korean government is probably guilty of crimes against humanity. Many governments which accuse North Korea have a less than perfect track record themselves, and they are willing to overlook despicable acts by their own allies, but even so the North Korean government is clearly evil.
North Korea also appears to be unstable. The government fosters a sense of external threat and crisis to rally the population in the face of internal failings, like their inability to feed the population.
Read about prospect theory. NK is clearly "operating in the domain of loss" which makes them prone to take risk. Or read about power transition theory. NK is a revisionist state under the terms of this theory.
But what about ...?
Pakistan is not exactly an upstanding democracy under the rule of law. Yet they got away with becoming a nuclear power. That might have been a mistake, assuming that there ever was a realistic option of stopping them. So far they didn't use their nukes. Why add another risky nation to the mix?
President Trump's tweets appear just as intemperate as those of Supreme Leader Kim. But the US are an established nuclear power and suggesting that they disarm is a bit impractical.
We should worry about a nuclear North Korea, but should we stop it?
Doing that would require a negotiated solution, a violent regime change, or sustained pressure to bring the regime to it's knees.
There seems to be little hope for a negotiated solution. Kim remembers Gaddafi, who got toppled by forces allied with the West after he gave his WMD away. And he remembers Ukraine, which gave her nukes away against solemn international security guarantees. For that matter, he sees how the US is treating the Iran nuclear deal. He wouldn't believe any Western promises. Likewise sanctions are difficult unless the North Koreans believe that there is a carrot as well as a stick.
A military disarmament won't be possible without a fight to the finish. It is just too difficult to find weapons hidden in tunnels in the mountains.