Dont confuse a beater bike with a junk bike. You need a nice enough frame and some good enough components.
MTB hubs are good, and dont stint on the BB and headset. Avoid thin-walled rims, they will wear out, and make sure they are built up properly. Winter commuter bikes take a hammering and need to be strong.
Full length fenders, which fit closely to the tyres will keep almost all the dirty water and road grime off you and the bike.

Water can conceal potholes and you want to avoid punctures, so use tyres a little wider than a racer. 28mm are standard for a fast commuter bike. Tyres and fenders need clearance at the frame and brakes.
You can use a cyclo-cross, light touring or touring bike, or a roadified MTB (but you cant get close fitting fenders). In the UK, some shops sell cheap winter training frames just for this purpose.