Both are used. Just earlier this month, a blogger wrote:
It’s easy to fall into the habit of saying that ‘everyone has the internet’ when what we all really mean when we say that is ‘everyone I know has the internet’. In actual fact two-thirds of people around the world don’t have the internet. Two very ambitious projects are underway to try to change that.
The word internet has entered the vernacular relatively recently, and people are still sifting out how to best use it. There was a time when it was treated like a proper noun (always capitalized, much like this columnist – or his publishing editor – seems to prefer); it now gets casually tossed around as a qualifier (as in, "We won't have internet access tomorrow.")
I don't think the dust has settled enough to say that one way is right and another is wrong, although there may be certain circles of people who favor one wording over another.
A Google search for
"I don't have internet" returns roughly 15,000,000 hits. Add the word the to the search –
"I don't have the internet" – and the number drops to about 4,000,000. That's significantly less, but not so few that I'd declare the wording "funny", or even "less natural."
I continue to see "internet" and "Internet", used both with and without the the. I don't think any of the permutations have fallen out of favor so much that I'd consider any of them "wrong," even if it's not written in the same way I would write it.