How do you keep a dog from going downstairs?

They are probably attempting to "claim" this space, and therefore attain a higher perceived status (above your kids) in their "pack." The best thing to do is keep the doors closed and let them know they are not allowed in the rooms. If you want them to be allowed in the rooms, you will have to keep a close eye on them and as soon as you see them begin to 'do their business,' make a sharp noise to startle them, yell "no" or "stop", and quickly move them outside or onto their piddle pad. This will have to be repeated many times before they get the idea.