U.S. alternative to airing cupboard
In addition to clothes driers, as explained by Jazzoctopus, there is also the fact that nearly every house in the U.S. has central heating, which keeps the entire house free from dampness, at least in the season when the heat is on. In summer, damp is not a great problem except in tropical places like Maryland, S. Carolina, Louisiana and a few others, which rival Calcutta and Central Africa for unbearable heat and humidity. Air conditioning solves this problem.