Seasons only have an influence on the catnip production of fields. The farmer job is not affected by the weather.
It is best to not change your kittens very much, because of the bonuses they'll get when they gather enough experience. See for example here or this question. It is not harmful if you've got a negative catnip production during winter, as long as there is enough produced and stored during spring/summer/autumn. So have enough fields and just a couple of farmers (I believe I've had 2 to 4 farmers, and as many fields as I could buy, and I had an overflow of catnip).
Edit: The default formula for the production is the following:
perTick = (building output * season modifier + job output * upgrades * happiness) * building % effects + automatic effects - consumption * consumption % effects
You see that the season modifier is only multiplied with building output, not with the job output.