That reminds me of Dubya picking Dick Cheney as his running mate; Cheney didn't scratch a demographic itch in the electorate but gave Bush a solid national security backstop that helped ease the IHOP-diplomacy-school meme that governors tend to get.
The same would be true of Rice; there isn't much of a voting block who would switch over to Romney because of Rice being on the ticket, since she appeals largely to the establishment Republican that was largely in Romney's pocket to begin with. She'd bring gravitas that most of the usual suspects on the VP list lack and be able to hit the ground running as an international surrogate for a President Romney.
However, she'll not shore up Romney's right flank, especially on social issues. Rice is a moderate Presbyterian and is not a culture warrior, which will make folks a bit nervous when Romney himself is a newly-minted social conservative, albeit seemingly earnestly so. Dubya had his evangelical Methodist background to offset Cheney's mainline Methodist persona; Romney doesn't quite have that luxury.
Is that enough to lose Romney support on the right? Most folks in that demographic will see the ticket as an improvement over the incumbents and vote accordingly. The folks who are already queasy about a Mormon in the White House might wind up pulling a Constitution Party trigger, but there will merely be a decrease in fervor with a MR-CR ticket.
Condi won't win over much of the black vote; she'd likely get a chilly reception at an NAACP gig as well, where quips about old Mormon racist theology would get swapped out for "House Negro." Nor will she win over too many women; a few RINOey types who hold Planned Parenthood fundraisers at the country club might come alongside but she would have little allure for the blue-collar moms that Sarah Palin could reach.
Romney could well do worse; a Rice pick would save money, as she could take the National Security director portfolio and save some payroll. However, it runs the risk of dispiriting the conservative base without doing much to win over swing voters.