What does a druid do on the winter solstice? That depends on the druid.
If you’re a Reconstructionist, you don’t do much. There isn’t a whole lot of evidence that the ancient druids did anything to celebrate the two solstices and equinoxes; their high holy days were the four cross-quarter holidays (Imbolc, Beltane, Lughnasadh, and Samhain). The solstices and equinoxes aren’t even marked in the Coligny calendar, for example, while there is evidence there for Samhain, Lughnasadh, and Beltane.
If you’re a Revivalist, you celebrate Alban Arthuan, the festival to honor King Arthur and the return of the light. Druids commonly gather in a sacred space and watch for the sun to rise, greeting it with the powerful “Awen” chant, and honoring it with ceremony. However, the Revivalists are not dogmatic, and traditions vary widely among them.
I have called myself Reconstructionist on this site several times, but honestly I’m not quite sure about that.