If Yellowstone decides to erupt, which is very unlikely, some people have looked into what would happen. They've found some very interesting things.
The Damage
According to this paper (which is free), North America will see a lot of ash in a super eruption. Some cities will get a lot of ash, such as Billings, Montana which is projected to get at least 1 m of ash. Other cities may see no ash, such as Portland or Flagstaff. The following map shows where ash has been found from previous eruptions.

The authors cites others about the negative effects of even a millimeter of ash, which can make rails too slick for trains, prevent electrical transformers from working, and even cause respiratory problems. Centimeter-thick ash covering the land would cause cattle and crop failure in addition to making many buildings potential unsafe and clogging streams and sewers.
Obviously, the ash cloud alone would kill people. Lowering the population of the world may or may not be "helpful" to humanity as a whole. The US still exports a lot of food to the world, which means that agriculture shutting down in the USA means a lot of hungry people elsewhere, especially in Europe and (surprising to me, at least) China. These are major economic and scientific centers of the world. I would guess that starving those places is generally not a good thing for humanity. I'm going out on a limb, but I think a Yellowstone eruption would cause us to lose many people, and the survivors would suffer a lower quality of life because of it.
The Benefits
These authors point out that the Pinatubo eruption of 1991 cooled the planet by ~1 degree C for several years. A Yellowstone super-eruption has spewed out about 280 ${km}^3$ about 1.3 million years ago. Pinatubo, in comparison, spewed forth only 5 ${km}^3$. Oh yes, Yellowstone would cool the planet by quite a bit; the USGS claims it would cool the planet for some years, possibly decades.
Killing off people would also drop the ability of humanity to produce greenhouse gases, just because the remaining people would have to pollute a lot more to compensate. This is unlikely to happen. A dramatic reduction in people may help us reach some of the carbon-emission targets. It may not fix the underlying problems with human-driven climate change, though, and remove some of the urgency of the change, which means this eruption would be a mixed blessing.
Back to That Paper
That first paper I mentioned ends with the statement that an eruption of this scale isn't likely. In fact, they say that any eruption from Yellowstone will be on a small scale, such as a new lava flow. This kind of tiny "eruption" has happened many times, but poses no real risk to the country or humanity as a whole.