What are some drug-free alternatives that can help minimize asthma symptoms?

Check with your doc: There are no real alternatives. Have allergy testing, especially cats and house mites, and avoid exposure if positive. Inhaled steroids are the standard of care.

Removing triggers: allergy induced asthma can be helped by removing or minimizing your triggers. Animal dander, dust mites, pollen, etc. should be addressed. Remove rugs, carpeting, cover mattresses with a plastic cover are all good starting points. You can get "allergy reports" for your area from the internet especially during pollen seasons.Of course don't leave yourself without a "rescue" inhaler. Good Luck