My wife has asthma and has been coughing. Should I get a dehumidifier or an air purifier?

Not humidifier. While dry air may make asthma worse, in time humidifiers become a mechanism for spreading mold through your house. Coughing at night could be associated with either dust in your home, allergies to blankets, or reflux. See your asthma specialist and let him/her know what her symptoms are and when they occur.
See your doctor. If she is having uncontrolled asthma, she needs to be on steroids, only your doctor can make that determination.