From the USA point of view there are two types of arguments. The first ones are those that state that the less nuclear capable countries there are, the better1:
Less nuclear warheads at risk of being captured/sold to rogue actors.
Non-nuclear countries are less of a worry if they become unstable.
Every foreign country, no matter how friendly, is at the least a competitor. And countries that currently side with you may change their stance in the future. The less leverage other countries have, the better. For example, one WWII ended the USA stopped helping the development of UK nuclear weapons.
And, the reasons for what North Korea may be viewed as an specially worrysome nuclear power are:
It is certainly not friendly to the USA.
It is technically in a state of war with one of USA's allies, and occasionally acts aggresively against it (including exchanging artillery fire).
It has publicly conducted illegal actions in other countries2 (kidnapped Japanese citizens, murder of Kim Jong Nam).
It is a totalitarian regime, which means less controls preventing a single individual deciding to use the nuclear weapons.
It has little foreign contact, which means both less leverage against it (has no foreign trade to lose if an embargo is placed) and less knowledge of its internal politics (what things are they interested in and which things they see as threatening).
1And yes, you can claim that is hypocrital for the country with most nuclear weapons to have this aim. Others can claim that, while this approach benefits the USA, it also benefits the rest of the country. We are looking at the reasons *from the USA point of view*.
2Here again, the USA has done that, too. Did I told you that these were the reasons from the USA point of view?