That depends, if you want to get to your destination in a hurry, or if you want to see some great sights along the way. All you need to do is watch the weather and drive accordingly. Even after a heavy snow, the River Road will be cleared the next day.
Obviously if you watch the news, you know that they also close the interstate highways when things get too bad. We have driven the River Road in the Winter and it was no problem at all. Be sure to call to check on your anticipated stops to make sure that the RV park is open and that you can get in to a parking place if there is any snow. Some RV parks do not clear the snow very quickly and others do.