My husband has esophageal cancer. My mother wants to come back from florida and smoke in our basement apartment. Is this safe? She says she will change the filters in the furnace once a month and add an exhaust fan an another air purifier. She also has a

Clearly . Clearly you are inquiring about the effects of second hand smoke. For the exhaust fan to be effective it would need to remove smoke directly from the room that she is in. Since your husband has a serious form of cancer, and it is your home; why not make a rule that there is no smoking in the home. Regardless of what special arrangments are made, if she is a heavy smoker it will permeate your basement apartment. If your mother is not willing to respect the rules in your home, you might take that into consideration regarding her desire to live in your home.
Safe . Safe for whom? It is certainly not safe for her. One would hope that she would see your husband's condition and be scared out of her habit. Nor is it safe for you and your husband because of the second-hand smoke issues that have already been raised. In fact, it strikes me as particularly insensitive--again because of your husband's cancer--that she would even ask.