A cough can be caused by many different causes - most not concerning, but some that indicate a need for medical attention. I would recommend seeing your primary doctor
if your cough persists or if you develop warning symptoms (fevers, change in your sputum color, shortness of breath).
To answer your question, humidity probably has little effect on your cough. A cough is a normal reflex that the body has developed to clear the airways. Any part of the airway can trigger the reflex, however deeper part of the airways (the trachea and bronchi) are the most sensitive and therefore the most likely to trigger the reflex. Air that enters the trachea is both humidified and warmed by the body before it reaches the deep airways. Therefore, regardless of the ambient air, the humidity of the air that reaches your airways is likely the same. It is true however, that the upper airway mucosa (like in the nose or the throat) can cause a cough and that the humidity from the outside can affect these areas - but this is probably less likely. In fact, dry air is probably more irritating the the airways then humid air. Therefore I doubt the humidity is truly exacerbating your cough.
Regardless, finding the cause of your cause is important. Infections like bronchitis
and pneumonia can cause a cough. Smoking is a common cause. The most common causes of a chronic cough (>8 weeks) are reflux disease (heartburn), asthma, and post nasal drip (or allergies).
I recommend seeing your primary care doctor