It's usually raining even before I start my morning commute, this time of year. I put on my rain gear, and ride.

I don't usually bother wiping my chain down at work, but I will when I get home in the afternoon. If it is a dry ride home after a wet morning, I'll usually give the chain a light spritz of <insert your lube/wax of choice> and that's about it. I generally clean/lube my chain about once/month in the summer, and about twice that in the winter.

Depending on the cold/wet, it might be worth investing in some good rain gear. I commute on a recumbent, so a poncho wouldn't really work for me.