There is pretty fast internet available in Barrow (T1 line), as cyber-infrastructure was built to support the science projects that NSF, NOAA, DOE, etc. fund there. However these routers are not free or public. The several hotels have internet though for their guests. You can also use the free public wi-fi in the library, which is next to the Heritage Center, and across the street from the supermarket. It seemed to be pretty fast when I used it there. There is also supposedly free wi-fi in the Alaska Airlines terminal, but it doesn't seem to work IME.