The long-running General Social Survey includes a 10 word vocabulary test, from which you can roughly estimate IQ over large enough sample sizes. (Of course, it's biased in favor of people who are smarter with words than with numbers or images.) Audacious Epigone looks up the average IQs of white voters for each Presidential candidate 1976-2004.
Presumably, Republican candidates' voters generally average higher IQs overall -- in the exit polls, GOP voters average higher incomes and very similar education levels to Democratic voters -- but all the heat on this issue of who is smartest is generated among white people. When white Democrats go on and on about how Democrats are smarter than Republicans, they aren't thinking about all the blacks who turned out to vote for Obama this year -- e.g., in California, where Obama got 61% of the vote but gay marriage, despite the best efforts of Hollywood, got only 48% -- which Hollywood has ever since been blaming on media domination by the Elders of Mormon). In the 2008 exit poll, there was virtually no difference in years of education claimed among Obama and McCain supporters when aggregated across all races.
No, white Democrats only care about being smarter than white Republicans.
Audacious's analysis found several things of interest. On an IQ scale where the white average is set at 100, all candidates's voters since 1976 have averaged over 100. Dumb people don't vote as much as smart people and undecided swing voters tend to be not very smart either. Thus, the losing candidate in six of the eight elections had a higher IQ set of voters than the winner. In other words, losers tended to wind up with his base of people smart enough to have a fairly consistent ideology, while winners picked up the people who don't think about politics much and motivated the people sympathetic to his party in the left half of the Bell Curve to remember to show up to vote.
It's kind of like Jay Leno vs. David Letterman. Dave pitches his show at viewers with a 105 IQ, while Jay aims his show at 100 (I'm making these numbers up but I wouldn't be surprised if they were pretty accurate). Jay gets bigger ratings.
Third party voters, with the exception of Perot's, tend to have high IQs.
Republican whites tended to have higher IQs than Democrats in the early years, and as late as 1996, Dole enjoyed a 0.6 point edge over Clinton, but by 2004, Kerry had opened up a 3.9 point gap over Bush.