What can I do to supress my asthmatic cough?

Treatment... Treatment for asthma includes bronchodilator and steroid inhalers, as well as other treatments depending on your symptoms. Asthma, characterized by reversible, recurrent bronchospasm and inflammation, can have cough as the major symptom of bronchospasm. In this case, the cough is treated just as wheezing is treated. So, by relieving bronchospasm and airway inflammation, the cough is treated!
Treatment. There are steps of treatment. You need to have your symptoms controlled. See your physician, and add treatments till they are minimized. Uncontrolled asthma is dangerous and progressive.