Would living in a basement or half underground apartment be a bad idea for someone with regular sinus & allergy issues? Potential for must/mold?

Depends. on the condition of the building, even if overground, environmental control is one of the important factors in controlling allergic conditions. Check the aaaai.org or acaai.org sites for more information on environmental conditions and impact on allergic conditions as asthma / rhiosinusitis, good luck.
Possibly. Only if it is damp and moldy. Lots of houses have perfectly dry and insulated basements with no mold. But knowing New York City- this probably isn't one...