A Wall Street Journal/NBC News survey released last week found that 42 percent of Americans believed Apple should cooperate with law enforcement officials to help them gain access to the locked phone, while 47 percent said Apple should not cooperate. Asked to weigh the need to monitor terrorists against the threat of violating privacy rights, the country was almost equally split, the survey found.

That finding may have seemed unlikely in the wake of terrorist attacks last year in Paris and San Bernardino. In December, eight in 10 people said in a New York Times/CBS News survey that it was somewhat or very likely that there would be a terrorist attack in the United States in the coming months. A CNN poll the same month found that 45 percent of Americans were somewhat or very worried that they or someone in their family would become a victim of terrorism.

But despite the fears about terrorism, the public’s concern about digital privacy is nearly universal. A Pew Research poll in 2014 found more than 90 percent of those surveyed felt that consumers had lost control over how their personal information was collected and used by companies.

The Apple case already seems to have garnered more public attention than the Snowden revelations about “metadata collection” and programs with code names like Prism and XKeyscore. The comedian John Oliver once mocked average Americans for failing to know whether Mr. Snowden was the WikiLeaks guy or the former N.S.A. contractor (he was the latter).

Now, people are beginning to understand that their smartphones are just the beginning. Smart televisions, Google cars, Nest thermostats and web-enabled Barbie dolls are next. The resolution of the legal fight between Apple and the government may help decide whether the information in those devices is really private, or whether the F.B.I. and the N.S.A. are entering a golden age of surveillance in which they have far more data available than they could have imagined 20 years ago.

“It’s an in-your-face proposition for lots more Americans than the Snowden revelation was,” said Lee Rainie, director of Internet, science and technology research at Pew Research Center.

Cindy Cohn, executive director of the Electronic Frontier Foundation, said: “Everyone gets at a really visceral level that you have a lot of really personal stuff on this device and if it gets stolen it’s really bad. They know that the same forces that work at trying to get access to sensitive stuff in the cloud are also at work attacking the phones.”