Can mold around the house cause my child to develop asthma?

Not develop. It can certainly aggravate someones existing asthma. Imost asthma is genetic.
Unclear. It is unclear if the presence of mold leads to asthma, but the right (or wrong, depending on your point of view) mold may lead to an asthma mimic - allergic bronchopulmonary aspergillosis (abpa).
It may trigger event. Mold by itself does not cause asthma. It may provide the irritant or allergic trigger to starting an asthma attack, but the underlying asthma condition was existed before exposure. Asthma has many minor forms and some major. The worst cases seem evident in childhood but some adults have a "cough only" varient they may never attribute to asthma.