In fact it is a right for businesses to choose who they want to serve. An airline that only flies costumed clients? 100% a right in this country. They can even decide who they consider costumed and who not.
* In research I discovered that it depends actually on the type of service. In California the Unruh Civil Rights Act catches all businesses; but federally only certain types of businesses must follow the discrimination rules. http://users.wfu.edu/zulick/341/civilrightsact1964.html
Airports are included in the Civil Rights Act because "it serves or offers to serve interstate travelers".