This area has been investigated (see enclosed abstracts for some examples). At least part of this adverse effect is a direct evidence of very high humidity on the airways, including a stimulus to cough. Recall that inhalation of high humidity air has been used for many years to induce coughing in order to obtain sputum specimens for analysis.
Individuals with asthma appear to be particularly affected by such inhalation of high humidity air. Also, the high humidity environments tend to have very high airborne levels of molds/fungi to which the individual may be allergic. In addition, dust mite proliferation is enhanced considerably when the relative humidity exceeds 50%; mite-sensitive individuals will be likely affected.
However, in my experience, asthmatic symptoms (particularly the "asthmatic cough") may also be worse in excessively dry environments (relative humidity below 15%). Therefore, I
have advised asthmatic patients to aim for a "happy medium" relative humidity in their homes, monitoring their home humidity regularly with a reliable gauge. Depending on the ambient relative humidity, dehumidifiers or humidifiers can be used judiciously to avoid extremes of relative humidity. I also advise asthmatics who appear to be very sensitive to high humidity to stay indoors in a humidity-controlled environment when the outdoor weather is very humid. If the individual is allergic to molds and/or dust mites, additional avoidance/control measures are indicated.