With the proliferation of wi-fi, and most locals and travellers carrying tablets and/or smartphones, the old-fashioned internet cafe is a dying breed in Denmark. There may be a couple catering to gamers and laptop-less travellers in the major cities, but public libraries are your best bet in mid-sized and small towns for free use of computers with internet access.
Libraries also have free wi-fi (you will generally need a code), as do many cafes and bars, and trains and buses. Wi-fi is ubiquitous in hotels and hostels and is usually free. Some hostels and hotels will offer a computer for guests to use, free or for a small charge.