It is at least asserted that air humidity affects health relevant environmental factors, but if you are interested in marketing for a humidifier or dehumidifier, you can freely pick arguments both in favour of dry and humid indoor environments:
Recent research shows that the absolute air humidity has a strong impact on the survival rate of air-born flue virus. Cold air (which is dry in absolute terms, although relatively humid) or dry air increase the flue virus' life span. Increasing the air humidity will reduce the risk of a flue infection.
Dry air on the other side, will however prevent other factors with negative health impact. Most fungi spores will only thrive under high humidity, so keeping a dry climate will avoid mold growth.
These are of course only two examples allowing opposite argumentation for a dry or humid climate. There are endless other examples of germs or e.g. pest carrying insects, which presence or amount can be controlled by manipulating air humidity.
Another issue is the vaporization of toxic solvents from building materials, which increases in dry air, but the "real" solution to that problem is probably to avoid such building materials completely instead of minimizing the risk by increasing air humidity.