Would dry air tend to make a person cough more if they have allergies or asthma than humidified air?

Usually: Dry (and cold) air is particularly bad for exercise induced asthma and that frequently causes cough. Swimming was always said to be better than running for asthma, but now with public pools so highly chlorinated, that also often makes asthma worse. For nasal allergies, the best way to humidify is to use saline nasal spray or flush.

Usually but: Dry air is generally an irritant to anyone, and makes asthma worse... But humidifiers generally are not the cure to this. It is very difficult to keep a humidifier from becoming a mold disperser.