It depends on how in depth you want to go. Since this is a kind of vague question, I'll just throw out some ideas.

You'll want particle effects, obviously. Some white flakes that'll blow from the horizon at the camera. You may also want some cloth physics to blow the player's clothes around. And you could also do a frost effect on the player's screen.

Kind of a hack, but you can alter the gravity of your physics, or use a summed vector of the wind direction and gravity to influence the player's movement. The downside to this is that it does not easily support local variations in the wind, if you want to have wind currents in certain parts of your map.

Similarly, have snow particles react to the gravity. They would be falling regardless of whether the air is calm or not, but of course, also add some air resistance to slow their downfall.