Are fern plants a good plant to keep indoors for cleaning air if your family has a lot of allergy problems?

Unfortunately no: Ferns are wonderful plants, but to really clean the air it takes a powerful air cleaner/purifier such as a hepa. This can remove up to 99% of airborne pollutants if chosen for the right size of room. Avoid ozone generating air cleaners. The moist dirt that live plants grow in can be a place for mold to grow--mold can be a significant indoor allergen.