If a visibly ill person is allowed to fly, is the airline liable if other passengers get sick?

No, the airline is not liable. For there to be liability, there must be a duty to take care--a legal duty, that is. Airlines are not under any such duty w/regard to colds or flus; it is not their responsibility to monitor the health of their passengers, and indeed, the airline would face breach of contract liability for refusing to fly people who've purchased tickets because they may have the flu.

Also, as a matter of causality--even if there had bee a duty on the airline's part, it would be impossible for you to show that you contracted an illness from that person on that flight, instead of from any other person or random stranger you come across in your life.