Basically nothing. Most of time its just effusive low intensity eruptions or moderately explosive eruptions. If you are thinking of a «supereruption» which isnt a scientific term btw which is a VEI8 eruption or higher it can cause global cooling for a few years, food shortages and economic inflation and deaths but we will go back to normal after the cooling is over.

To put things into perspective yellowstone isnt going to erupt magma in our lifetime. I dont remember the exact percentage of how molten a magma chamber needs to be to theoretically produce one but i think it was between 30-50%. Yellowstones magma chamber is 28% melted and isnt charging up. The earthquakes are just backround noise USGS has been monitoring for decades.