Snow is crazy hard.

if you want to do it in post, you'll have to composite multiple layers of snow, at different scales, moving at different speeds. That way it looks like a dense amount of snow in the air.

If you want to do it on set, its equally as hard, but maybe more fun? It depends what kind of shot you're doing.

Your first choice is to get a few sheets of glass and actually make bits of snow (I think maybe some lumpy flour, if you throw it at the glass when the glass has been wet? That way it'll dry onto the glass without giving you too much trouble). You make three or four panes of this, and move them every frame, positioned at different distances from the camera. Not only do you get snow, with realistic blur, but you also get depth of snow, which will make your characters look like they're really in a snow storm. Just make sure your lights are coming from behind the glass, or you'll get some terrible reflections.

Your other option is to shoot your scene without snow, but sprinkle flour (because icing sugar can be a little harder to clean up?) onto the entire set using a sieve (or a strainer, just shake it gently). So you're applying a little more snow every frame, and also maybe blow it gently every frame to make it move. Shoot that, without worrying about the falling snow. Then when you're done, either film (liveaction) a clean frame of your set while you use the sieve to sprinkle snow down in front of the camera, or you replay the clip on a screen and film it (liveaction) while you do the same thing.

Hope that helps