There are many reasons why fewer countries armed with nukes are a good thing (both for the US and the world at large).
First, limiting control of something help prevents it's use. Do I entirely agree with this? No, but at the same time I'd rather not have countries nuking one another in any war they get into (or even have that option for a commander to go "fuck it" as a last ditch effort).
Second, limiting control also prevents accidental use. There have been many times when the US and Russia have almost accidently launched. Be it from faulty information, computer errors, or something else. Keep the countries that have this option small, and it's less chance for accidental launches.
Third, MAD. Take a country like NK launching against SK for example (assuming both had nukes along with Japan). NK launches against SK, and you have SK going "OH FUCK" and launching their nukes against NK. Japan detects launches, and they might launch just in case it's against them. China gets involved, and within 30 minutes you have a runaway MAD case going.
Fourth, look at what happens when a nuclear country falls. When Russia fell, they had a lot of nuclear weapons/material that could be sold on the black market (if any were, who knows but that possibility was there). If you allow nations like NK, Iraq, or any other "unstable" nation to have them, you increase the chances that they fall into "the wrong hands."
Fifth, the only "good" that can come out of more countries being nuclear armed is that more countries are able to have that deterrant. Unfortunately, if Saddam had nuclear weapons I have little doubt that he (or commanding officers) wouldn't have launched when the US invaded during the Iraq war. I don't want that to happen.
Sixth, you say control of it is bad and we shouldn't prevent countries from developing nuclear capabilities (along the lines of "if everybody has them"). Well take gun control for example. I believe that giving everybody guns is the right choice, because it allows them to protect themself. Unfortunately, there are still crazy people out there who don't think "oh hey he has a gun so I won't do something" and do it anyways. This is especially true when you get fanatical beliefs/ideologies. Do you have any question that men like Hitler wouldn't have used nuclear weapons if they had them available? I don't.
Seventh, the nuclear genie has been let out. We can't put it back in, but we can help limit/slow the expansion of it. Nuclear weapons have no "counter", and are the deadliest weapons we have ever created (except maybe some biological stuff maybe). Previously, all weapons that were developed (with exceptions of biological/chemical, but those are very easy to come by comparatively) had a counter to them, or couldn't be the "end all" to a war. Nuclear can, so I wouldn't want lots of offensively armed countries all going "don't fuck with me or else."
Eighth, what benefit does allowing any country to go nuclear provide? More security? Nope, especially when countries like NK have them that aren't "stable". It actually helps to destabilize a region when a country goes nuclear. All the non-nuclear nearby countries go "oh shit, what should we do?" and it can risk war. I think we can agree wars are not good for the world.
Ninth, the proliferation treaty many countries signed. SK being one of them. Now you want it? Make up your mind.
Tenth, the resources required to get into the nuclear club is quite high. Once there the cost of ownership isn't low. So why spend lots of money, to research, develop, build, and then maintain a purely offensive weapon that it just takes one mistake to wipe a country off the map (by launching, accidently or not, a nuke).
Eleventh, it's time for lunch.
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