So the US did not do any nuclear testing in Arizona. You are probably thinking of its testing in New Mexico (in 1945) or Nevada (from 1951-1992).

As for why it chose Nevada for its main continental test site, there were many reasons. The #1 reason, though, was that it was convenient to use. They already had a Pacific test site, but it was very hard to truck out there, set up tests, get diagnostic information, get things back, etc. The weather was mercurial. They wanted a continental test site that would allow them to test relatively small bombs (compared to the Pacific ones), test them rapidly, and work in close conjunction with their laboratories in New Mexico and later California. They also already owned the land (it was a bombing range prior to being an nuclear testing range), and it was sparsely populated (with the exception of nearby Las Vegas, but it was felt that with proper attention to wind direction any threat there could be eliminated).

This is not to say that it was entirely unpopulated, to be sure, and there were a lot of people living downwind of those bombs who are today considered victims of the nuclear testing. But the reasoning in the 1950s was that this was not much radiation for them to be exposed to, and that national security required breaking some eggs to make an omelet.

New Mexico was chosen for the first test largely because of its proximity to the laboratory at Los Alamos. Other sites were, however, considered. One site in California, for example, was nixed because the head of the Manhattan Project learned he would have to get permission from General Patton, and he loathed Patton, so he crossed it off the list.

As for Alaska — the US did test several weapons in Alaska in the 1970s, underground. Logistically it was difficult because of the remoteness of the test area, the notoriously unfavorable weather, and the fact that the area is very seismically active. The tests were extremely controversial, but carried out in the face of considerable public protest and local opposition.