I think it might also be possible you are talking about the "No Fly List", so in case you are, I'll gear my answer towards that. This is the list maintained by the federal government that refuses some people, Americans and foreign, access to air travel into, out of, or within the United States.
The civil rights issues here are a little tricky. Americans have no right to get on an airplane, and government has the authority to regulate air travel and safety. The lack of transparency with the No Fly List is problematic, though.
First, you are much more likely to be on the list if you are Muslim or Arab-American, or have a Muslim sounding name. So it is a form of racial and ethnic as well as religious profiling.
Second, once a person is on the list, not only can't they fly on an airplane, but anyone else with that same name cannot either. That is, the list does not distinguish between different people with the same name. This is how some infants and teens have been denied air travel.
Third, the criteria for determining whether you are on the list are, well, classified Top Secret. The government won't tell you how you got on the list. There have been some cases where a person called to inquire if they were on the list before they attempted air travel, and their names were subsequently put on the list. Want to know how to get your name off of the list? That's classified too.
So the main issue of civil rights here is the lack of due process any citizen or visitor is accorded with regards to the No Fly List.
(And if you do mean No Fly List instead of No Fly Zone, maybe message us or edit your question to reflect that. Thank you)
The question orignially read:
"How does the No Fly Zone policy in the United States affect civil rights?"
I assume that you are using this term to refer to areas in the US where pilots are not allowed to fly, usually for security reasons. The more usual type of no-fly zone (in a foreign country, set up to protect the people of a given area from air attacks by their own government) does not seem to have a lot to do with civil rights.
I also assume that you are using the term "civil rights" to mean something like the right to travel freely. The more usual usage of the term (which refers to protections of rights of minority groups) does not seem to apply here.
If these assumptions are correct, I still do not think that rights are impacted by no-fly zones in any substantial way. Americans have a generally accepted to move about the country without any limitations. However, this right is subject to many restrictions having to do with private property or government installations. The limitations placed on flights in no-fly zones simply do not seem to be very important. People's freedom of movement is not materially harmed by being unable to fly airplanes over the White House or Capitol building. This may be a minor infringement on our rights, but it is not anything that should be a serious concern.
I'm sorry, I should have typed "no fly list"!