Very simple. Have like a temperature meter, when you are in harsh climates like snow or desert biomes, and when it drops to zero or gets too high, you start taking damage slowly or you run out of stamina in cold.
Can have clothing/armor items that increase your resistance to these climates.
Freezing cold waters should do damage, like lava would too.
The idea sort of came from Starbound. All snow/desert doesnt have to be damaging, but could have it in the form of the occasional blizzard or sandstorm that occurs randomly, forcing players to take temporary shelter.