Should the word "internet" be capitalized ?

Not in this blog.  But apparently there are different styles, as reported in the Oxford Dictionaries blog:

The latest salvo in the capitalization wars came from the Associated Press Stylebook, which announced that as of June 1, the AP’s style will stipulate that internet and web (with reference to the World Wide Web) should be lowercased. The AP’s is not the first style guide to insist on downcasing internet; many other publications prefer the lowercase form as well. And yet, attentive readers may notice that the headword form in Oxford Dictionaries continues to harbor a capital I (at least for now).

The reason Oxford has retained the capital I is simple: evidence. Our research samples continue to show that the capitalized form of the word is slightly more common. Over the past few years, the proportion of evidence for the two forms in our monitor corpus has remained relatively steady from month to month, with capital-I Internet accounting for about 54% of all examples. Dictionaries are lagging indicators of language change, waiting for new usages to become settled before recording them, and this particular change is still underway...

Why did Internet come to be capitalized in the first place? In fact, the earliest use of the word, cited in the Oxford English Dictionary from 1974, was with a lowercase i. Initially, there were many internets—the word was used to refer to any computer network comprising or connecting a number of smaller networks; it later came to refer specifically to the global network we know today, which was distinguished as “the Internet” as opposed to “an internet”. It isn’t uncommon for words to take on a capitalized form in a particular meaning regarded as a proper noun (for instance, Americans style the foundational document of their federal government as “the Constitution”); however, in the case of “the Internet”, by the mid-1990s the original need for disambiguation was largely obsolete, and the capitalization convention began to strike some writers and editors as unnecessary, dated, and aesthetically unappealing.

We are now in the midst of stylistic change with respect to the capitalization of Internet, but the process is proceeding in patchwork fashion and is far from complete.