This article in MediaPost summarizes the broadcasters' case against Aereo this way:

Calling Aereo a “direct assault” on the broadcast industry's business model, a coalition of TV companies indicated in court papers that Aereo's continued existence could mean the end of free over-the-air television.

In my reading of the Constitution, I see neither a right to free TV nor protections for an existing business model (snark over). 

But what this line of reasoning does show is how fundamentally the TV business is changing after decades of stasis and how difficult that is for an industry that has grown so comfortable — and profitable with the status quo.

Are content creators entitled to fair compensation for their work? Of course. Does that mean that a business model that requires every cable and satellite subscriber to pay for every channel carried on the system, whether they watch it or not, deserves Constitutional protection? Sorry, I don't think so. But, of course, the Supreme Court will have the last word. 

This is a topic I'm covering deeply these days. I invite you to read some of my reports and posts to get up to speed on the forces driving change in the TV business:

Digital Disruption Rattles the TV Ad Market

How Software is Eating Video Ads and, Soon, TV

How Online Video Will Challenge DVRs' Role

Is Aereo About TV Or The Cloud?