The Solar Storm of 1859 (a.k.a. the Carrington Event), which pushed both sets of polar aurorae down to tropical latitudes and caused telegraphs to fail all over the world (while some worked via induced current even after being disconnected from their batteries). If such a storm happened today, it could be devastating to the modern world.
Red dwarfs are smaller and dimmer than our Sun, meaning their habitable zones are closer, and so any planets in them are likely to be tidally locked (i.e. the planet would rotate once every time it orbited its star). Tidally locked planets are unlikely to have strong magnetic fields to protect them from solar flares. On top of that, red dwarfs (especially young ones) are very prone to flaring, and can produce BIG flares that put our Sun's flares to shame.