Gilmore v. Gonzales (2006) sort of dealt with this. Dude wanted to fly, refused to present ID, refused the heightened security check, was told he couldn't fly. He sued, it went to the Supreme Court. The Supreme Court ruled that because the heightened security check was an option, the claimant's freedom of movement was not restricted.

There's a bunch of case law about freedom of movement, which is pretty radically protected in the United States. Not only can the federal government not put up unreasonable impediments to interstate travel, individual states and even private companies can't, either. Since American's aren't required to have ID (interestingly, this might be more political than legal, I can't find any case law enshrining the right to not have ID, just a lot of public outcry and backpedaling any time it's suggested), requiring one to travel interstate would be a significant impediment.