The article "Democrats gain edge on border security" directs our attention to this recent poll:

One national tracking survey by the nonpartisan Public Agenda Confidence in U.S. Foreign Policy Index ranks stemming the flow of illegal immigration as a top international relations concern for the American public.

Asked by the survey to rank proposals to improve U.S. security, respondents rated the tightening of immigration second only to improving U.S. intelligence operations.

Three-quarters of the 1,004 Americans surveyed for the index from June 1 to June 13 gave the United States a "C" grade or worse in protecting the borders, with nearly one-quarter handing out an "F" grade. Fifty-eight percent said tighter controls on immigration would strengthen national security "a great deal." The margin of error for the survey was plus or minus 3 percentage points.

If you download the PDF at the second link, you'll see that only one other question got a greater "a great deal" percentage: "Improving the effectiveness of our intelligence operations". The statement "it may be too easy for illegal immigrants to come into the country" generated 42% who "worry a lot" and 31% who "worry somewhat". Only 27% "don't worry". Only 2% named immigration as the "most important problem facing the United States in its dealings with the rest of the world". It probably isn't, so it would have been a bit more complicated but a more complete picture if people had been asked to rank the top five issues or so. The AZ Republic article also contains this:

Luis Miranda, a Democratic National Committee spokesman, stopped short of acknowledging there is a concerted strategy by the party and its candidates to seize on immigration as a potential wedge issue.

But he said the Bush administration and Republicans who control both chambers of Congress "haven't addressed the need for comprehensive immigration reform . . . leaving states in a bind."