Could a block basement with an old cistern cause respiratory problems?

Yes: Block basements and old cisterns are often associated with mold growth. Exposure to molds has been associated with a variety of respiratory disorders and should be avoided. Mold grows best in wet areas so it is important to keep basements dry and avoid household water leaks. If visible mold is present in a house it should be removed/treated by an expert.

Yes: Anywhere that is moist and dark is a risk for mold to grow. The spores (sort of like a dust) coming from mold can be inhaled and cause several kinds of lung disease as well as allergy for the people living there. For the most safety, the area should be inspected and, if needed, cleaned up and dried if possible. It is especially important if anyone in the house has any kind of breathing problem.