The US has seen protests at airports, for example the recent protests against Executive Order 13769. The number of protesters are probably not comparable though. While I can't find numbers on protesters at Hong Kong Airport, the aforelinked Wikipedia page lists over 2000 protestors at JFK.
Those protests were allowed, as far as I can see. Obviously, a few thousand people isn't that many for a large airport like JFK and the protesters didn't seem to try to block the airport.
The Wikipedia page does give us an insight into how authorities may deal with a few protesters having been arrested. From the linked Los Angeles Times:
At 10 p.m., two people were arrested for blocking the roadway, according to LAX police Officer Rob Pedregon. The two were cited and released.
As for a protest the scale of Hong Kong, I can speculate. With a protest of that size, the authorities will have difficulty guaranteeing the safety of passengers and staff. Especially when you're with many people, you can easily overrun secured areas of an airport as airports aren't used to dealing with that.
Depending on the situation (that many don't just protest for fun, at risk of being arrested) it will probably be condemned by many and the authorities will work on clearing the airport (e.g. by detaining and citing protesters).