Yes musty smells in a house or building are dangerous to your health because the musty smell indicates that there is mold growth somewhere in the house or building.
When you have a musty smell in the house or other place you should find where the musty smell is coming from and get rid of it.
An aunt of mine lived in a musty house and she developed a lot of breathing issues and then developed lung disease and lung cancer and eventually died at age 60 years old.
She never smoked in her life and just would not get rid of that musty smell in her house and I could never stand to be over at her house.
So if you have a musty smell in your home you need to find the source of the musty smell and get rid of the mold and musty smell from your house.
The musty smell could be mold on carpet, mold on drywall, mold on the flooring where you cannot see it or mold in the ceiling etc.
Also use a dehumidifier to keep your house dry of moisture during humid days.