This seems due to messy implementation causing some arriving travelers to get held in transit, unable to leave the airport. The suddenness of the rule change has also created a set of extremely vivid travel nightmare scenarios--people on vacation unable to return, employees and students stranded attending conferences and professional events, émigrés moving to the US now without a place to stay.
To say nothing of the long-term, structural effects of this policy on US security, standing and vibrancy, I think it's fair to say that coordinating protests around airports is a somewhat novel and interesting tactic.
"If they can't fly, no one can"
I.
Protests typically emerge in a somewhat complex, bottom-up manner--a mix of both explicit and implicit coordination. As a result, most protests happen in "logical" places that sound like places most protests would happen. City centers, town squares, police precinct buildings, Trump hotels and property. Sometimes, however, activists coordinate around unlikely locations; think about the recently-rediscovered innovation of seizing major highways.
The highway shutdown strategy is fascinating in part because it reveals how many US cities lack a truly universally-recognized "central plaza" that ordinarily serves as the dominant coordination point.
But it's also effective because it hurts random people who are just trying to go about their business. This may seem like a bug instead of a feature--those motorists aren't to blame for whatever injustice is at hand, after all. But in addition to raising the probability of making the news, randomly-targeted protests strike at everyone in a probabilistic sense. Interfluidity put it best:
In small matters, the fact that people will bear disproportionate costs to protest small ripoffs is essential to the integrity of everyday commerce. In larger affairs, the human propensity to altruistic punishment means we all bear costs of perceived injustice, we all have a stake in finding some mix of society and legitimating ideology under which outcomes are perceived as broadly right.
II.
So what about airports? Given the protestor goals, it seems to me that the attempted activist shutdown (or slowdown) of major airports has stumbled into a fairly smart approach. For many reasons, the psychology of air travel looms large in the public consciousness. Think about it--people are so weird about air travel! Everyone has their horror-story of missing flights, getting stuck, losing a passport, etc. etc.
The incredible vividness of the stories of people getting screwed by Trump's travel ban should easily trigger news consumers' flight anxiety and travel disaster porn mental subroutines. This seems likely to increase total empathy relative to the losers from other, more abstract policies, perhaps increasing the political pushback.
A second reason why protesting at airports might be an effective tactic is that federal politicians spend a massive amount of time in airports, and thus might be personally more exposed to disturbances targeted there. I am reminded of an old article by Alex Pareene about how a budget standoff quickly resolved after it started affecting air travel convenience:
“Shuttle flights between Washington and New York were running 60 to 90 minutes late,” the Times reports. Do you know who takes weekday shuttle flights between Washington and New York? People who think they are too important for the train, let alone the bus. People Congress listens to. (People Congress is, also.)
We'll see what happens.