You'll need your doctor's help. Asthma can be difficult to diagnose in children younger than 2, because conditions other than asthma can cause wheezing or wheezing-like sounds. In fact, viral respiratory infections are probably the most common cause of wheezing in babies.
However, if your baby coughs frequently and has allergies or eczema, and your family has a history of asthma and allergies or eczema (especially if you and your partner both have them), there's a good chance that your baby has asthma. His symptoms are likely to be worse at night.
Your child's doctor will examine your baby and take a careful history, including a family health history, to help make a diagnosis.