How does having any type of relationship with the North Korean government advance the interests of the United States government?
Even if the USA would like to have nothing to do with North Korea, North Korea's actions have an influence on other countries in the region, especially with some of the USA's closest allies (South Korea and Japan), and the world, at large.
While the USA's formal relationship is as limited as is possible, they can't actually pretend North Korea doesn't exist because they do exist, and, as stated before, their actions have impacts.
In order to try and mitigate negative impacts, the USA has to be able to at least try and have some kind of impact or influence on North Korea's behavior, which requires some level of interaction and engagement.
The USA engages North Korea for a few reasons:
The Korean question is also an American matter
Historically, the USA and the USSR somehow created South and North Koreas. Precisely, the current regime is an outcome or a relic of the Cold War. During WW2, Korea was occupied by Japan. After the war, the country was divided in two, the Soviet Union occupying the North and the USA occupying the South. In the North, communist resistant against Japanese occupancy Kim Il-Sung (grandfather of the current North Korean leader Kim Jeong-Eun) became the leader of the Democratic People's Republic of Korea, whereas the South Koreans got Syngman Rhee, a fervent anti-communist, as their first President. The Korean war, involving both Koreas and their allies, started in 1950 and though an armistice was signed in 1953, the war never officially ceased by a peace treaty. In addition to Wikipedia, Slate has a nice article stating that
Article IV of the Armistice recommends that "the governments concerned on both sides" convene a conference within three months of signing to organize the withdrawal of foreign forces from the peninsula and settle the "Korea question".
North Korea threatens the USA
Of course, as the Slate article states, the US Congress never declared war against North Korea, so the successive government could have considered that the war was de facto over, and decided to ignore the "North Korean troll". The problem is that North Korea directly threatened to strike the USA's overseas territory (Guam), and the mainland. It also threatens the USA's allies in the region, as explained in @PoloHoleSet's answer.
Edit(04/12/2017): North Korea recently tested a new ICBM missile that is said to be able to reach the American mainland, and could be deployed in 2018. See this article.
North Korea is a proxy
The USA is currently in a power struggle with two other powerful and influential nations: Russia and China. In particular, China is known to be the strongest allies of North Korea. The American involvement in the world's affairs is contested by these two superpowers, so withdrawing from the Korean question would mean acknowledging a defeat in the competition for world leadership.
A traditional conflict
This argument was developed by French writer Yann Moix, who visited North Korea a few times, and is preparing a book on the country. You can watch him explain it in this video (in French, subscription required). I am not sure I agree with it, but I think to it is interesting.
His point is that Donald Trump is aggressively engaging North Korea because it is a conflict that he can understand. Contrary to traditional conflicts that involved nations, modern conflicts involve non-clearly determined organization (ISIS, Al-Qaeda,...) without clear geographic localization, as well as unconventional weapons and strategies (cyber-attacks,...). North Korea has the advantage of having a territory with traditional objectives (cities, missile launch pads,...) that one could strike, and with a conventional weaponry.
North and South Korea are still technically at war
There was no treaty or real resolution, only an armistice. If war were to break out for any reason, Seoul (with about 10 million Koreans in easy striking distance of North Korea) would be the first casualty. It's a daunting fact that has definitely played a role in the US diplomacy (emphasis mine)
Things were so serious that on June 15, 1994, Clinton called Defense Secretary William Perry and Joint Chiefs of Staff Chairman John Shalikashvili into his office. They offered the president three options. Clinton could push for tougher United Nations sanctions (risky, because the head of the North Korea delegation said that sanctions would be seen as an act of war). He could move 10,000 more troops, long-range bombers and carrier battle groups to Korea in case a strike would be necessary.
Or, he could authorize a preemptive strike. The plan, known as “Op plan 5027,” would have sent cruise missiles and F-117 stealth fighters to strike the small nuclear reactor in Yongbyon. The hope was that the action would make it impossible for North Korea to turn raw material into bombs.
The Clinton administration felt confident that it could successfully fulfill that mission. But it also knew there would almost certainly be collateral damage. North Korea would have likely deployed its weapons against South Korea. “I believe it would have resulted almost certainly in war,” Robert Gallucci, the State Department's point man on Korea in 1994, told CNN in 1999. At the time, analysts estimated that 1 million people would die.
So we keep our diplomatic channels open. No president wants that kind of carnage under their watch.