The growing popularity of streaming may threaten ESPN, which is for now one of the most expensive parts of cable subscriptions, said Bernstein senior analyst Todd Juenger.

"ESPN is an extraordinarily powerful brand for all those people out there, and there are lots, who care about sports," Juenger said to CNBC.

"But the fact is not everybody cares about sports, and we think there's tens of millions of households who, for a variety of reasons, are still paying today for ESPN who don't really want the product. We don't think there's anything Disney can do to compel those types of people to keep paying for ESPN over time."

Streaming platforms such as Hulu and YouTube, as well as other companies such as AT&T, Dish and Sony, have given viewers more options outside cable and satellite. Theoretically, they'll also allow consumers to pick which networks they want to pay for instead of buying the entire cable bundle. People who don't care about sports can skip ESPN, which carries some of the highest carriage fees, the portion of the subscription cost that goes to the network.

Disney reported its quarterly earnings on Tuesday after the bell. Though the company was just shy of Thomson Reuters analyst estimates on revenue and beat on earnings per share, stock declined about 2 percent due to ESPN cutting into operating income.

CEO Bob Iger told CNBC that subscribers are leaving ESPN, but emphasized the company is making moves to make sure the network is digital friendly.

"We will eventually be in a direct consumer business for ESPN products, and hope to launch one of our brand technology platforms before the end of year ," he said to CNBC. "We are actually confident in ESPN's future."

For now, ESPN may be safe. Most of the major streaming plans all include ESPN, and Iger said on a call with investors the company plans to include ESPN on all the services.

"It's somewhat comical in a way," Bernstein's Juenger said. "One would think that you would see all sorts of varieties of the types of packages, offering things to different consumers. Strangely enough, what's launched kind of looks the same."

As a result, consumers aren't snapping up subscriptions to these live TV services.

"So far, they don't seem to be doing much of anything we can see to bring incremental paying subscribers into the system," Juenger said.

And Disney is still well positioned as a media company, especially with its movie studios and theme parks doing well, he pointed out. However, its cable networks are hurting and advertising demand is slowing for most media companies, he said.