It's well-known that the Internet's impact on everyday life is hard to measure in economic terms, because an awful lot of what people do on the internet is free. But Internet access—what I buy from Comcast at home and from AT&T for my phone—is certainly measurable economic activity. And Michael Mandel notes that according to the Bureau of Economic Analysis it's steadily on the decline.
Note that the BEA isn't saying that nominal spending on Internet access has fallen, they're saying that real consumption is falling which certainly doesn't comport with any normal observation about how people are living their lives.
The way Mandel puts this, the BEA is "undercounting the strength of consumer data consumption." I think that may be too strong a conclusion. What I'd say is that it's a reminder of the limits of this kind of real GDP data. In a lot of ways it makes more sense to look at nominal data—how much are we spending on Internet access—and then to look at quantities consumed. How many houses have broadband connections and how fast are they? How many smartphones are out there and how much do people use them? What I think you'd see is that the relative price of Internet access compares to other goods and services (rent, health care, food gasoline) is falling faster than quantities consumed are rising, so Internet access constitutes a "shrinking" sector even though people are actually doing more of it.