You could just teach the dogs to wipe their paws on the mats before climbing the steps.
Seriously though, The carpet will get wet if it gets rained on and when getting rinsed off. We are particular about tracking debris inside and will place an old towel on the steps if they are wet and also right inside the doorway. But the carpet will dry fairly quickly. About every other year, I'll take them off and clean the steps and add a coat of paint. But rust is very minimal and the steps will break before rust will ever be a problem.
You folks up north may have a issue with the road salt and may have to paint more often.
One of the first things I bought after buying the TT.