I think they've had pretty profound effects at times, though their exact timing and effect have been debated hotly among archaeologists, paleoanthropologists, and historians. There has been an ongoing debate about the Santorini explosion and its role in ending the Bronze-Age "Minoan" and Mycenean civilizations remembered in legend. There had also been a hypothesis (fraught with some timing issues that have limited its acceptance) that across the eastern Mediterranean it could have caused the plagues of the Exodus in which the Israelites escaped Egyptian slavery and genocide.
Large volcanic eruptions often cause yearslong attenuation of sunlight and cooling episodes that can lead to famines and glaciation.
Once Homo Sapiens was on the scene, such eruptions were thought to have prompted mass migrations worldwide as populations sought new food supplies. There's a theory that the Toba explosion in Indonesia (where some of the world's most active and powerful volcanoes are located) c. 74,000 yrs ago, and the ensuing estimated 6-year volcanic winter, led to some of the population ancestral to the Khoi-San (aka Bushmen) peoples of southern Africa to start migrating around the shores of the Indian Ocean and leaving various populations along the way such as the insular folk of the Andaman Islands, the "Negrito" peoples of the Philippines and other southern Pacific countries, the ancestors of New Guinea highlanders who established an agricultural civilization, and the Australian aborigines (though how they reached these places where there have never been land bridges to the mainland is still debated).
Michael discussed the Toba explosion and its theorized effect on the course of human development in this post.
We can expect major supervolcano explosions to have even more devastating effects were they to happen, given modern densely populated civilization inhabiting their very doorsteps. The Yellowstone supervolcano going off at full potential could devastate at least a third of the continental US and Canada directly or due to heavy ashfall, and set off a worldwide yearslong volcanic winter. The last major eruptions in that region long predated known human habitation of the Americas. I read a theory that its eruptive potential is held in check by the cooling it receives from boiling off some of the Yellowstone River. If the river were to dry up and fail for whatever reason, the building heat might bring on a devastating eruption.