What to do if my 3 week old baby been coughing a lot?

See the doctor. Its important that your baby's pediatrician check your baby to make sure there is no pneumonia, wheezing or other problem causing this cough.
See your doctor: A cough in a 3 year old infant is always something you should have evaluated. The age and issues of cough necessitate a thorough history and physical exam. My best to you and your baby!