It would probably be much like "the year without a summer" in 1816 after the massive eruption of Tambora in 1815. It wasn't a global catastrophe, but there were some crop failures and it was colder than usual.
On the other hand, Frankenstein was written as an indirect result. Byron, Keats, and Shelley were friends and they all got together at Shelley's house. Because of the bad weather, they stayed inside and decided to have a contest to see who could write the best scary story. Shelley's wife, Mary, joined in. She wrote Frankenstein. I don't think anyone remembers the stories that were written by the three famous writers, only the one by Mary Shelley. Just a little trivia there.
Anyway no, it wouldn't be a global catastrophe, but it would re regionally devastating. The good news is that there is no reason to believe that Yellowstone will have a "super eruption" within the next 90,000 years or so, if ever.