I would assume that Sarah Palin is about as smart as her erstwhile opponent, Vice President-Elect Joe Biden, a man who has gone through life with a giant chip on his shoulder about his IQ. Whether that's smart enough to be President, I'll leave up to you.
On the other hand, I'm sure Biden would beat Palin if they took a current events quiz on foreign affairs.
The chief answer is obvious, which means that the mainstream discourse is oblivious to it. Governor Palin is a lady. Specifically, she's a mom, a mom with a whole bunch of kids. If you aren't a mom, it's hard to grasp just how much more interesting your family, and the community they live in, is than the Law of the Sea Conference or the Tbilisi pipeline. In contrast, Mr. Biden, who isn't a mom, has time on his hands for paying attention to stuff like that because Mrs. Biden worries about the important things for him.