Nuclear weapons are fairly basic devices, easy to make. It is foolish to try to stop nations from building their own, the attempt will only damage relations. Every nation that has nukes, Pakistan included, has been very responsible in not using them for idle threats, and no one has used a nuke in combat except the US. The risk is not that every country could have a bomb, but that terrorists could get bombs. A terrorist nuke set off in Times Square or DC would be a world-changing event. Rather than alienate nations by trying to stop them from acquiring weapons, we should be building a solid international community determined to deny nuclear weapons from terrorists.
North Korea has them, and I would not call them responsible in the least.
South Korea does not have them, because they have an alliance with the U.S., thus they have no need of them. North Korea has an alliance with China, thus they do not need them, unless for destabilizing the region.