Many American style guides recommend capitalizing the first letter of Internet, and most major American publications (as well as many Canadian ones) do so. Outside North America, internet is rarely capitalized.
The non-U.S. approach makes more sense. There is no good reason to capitalize internet. The convention in English is to capitalize the first letters of proper nouns, which are the official names of people, places, objects, or events. The internet is none of these. It was originally capitalized to differentiate the Internet (the global network that anyone can access) from an internet (any network of interconnected computers), but in common usage this distinction is now irrelevant. Internet is now just a generic term for the communication medium.
The capitalized Internet still prevails in many major American and Canadian publications—for example:
Performance art, in the broadest sense, has existed since long before the Internet. [New York Times]
David Cameron appears to have had a change of heart on the subject of Internet freedom. [Globe and Mail]
A wastewater treatment plant worker’s photos of a ‘weird-looking’ creature go viral on the Internet. [Los Angeles Times]
Elsewhere (and in a few North American publications), the lowercased internet is more common—for example:
It may sound like the stuff of science fiction but these cyber cows are at the forefront of the evolution of the internet. [Guardian]
Spending too much time on the internet and mobile phones is an increasing problem in marriage and relationships. [Irish Times]
These days, the internet provides us with thousands, if not millions of outlets for news and opinions, catering to the most niche of tastes. [Sydney Morning Herald]