This is a surprising choice, and she's such an unknown that anything could happen. But a couple of thoughts:
1. If Obama wants to turn off yet more women, he can try to play this 'scandal' but I think it will backfire.
The trooper involved was not just in a custody dispute with Palin's sister, but had made threats to kill at least one member of her family and had tasered an 11-year old relative. He also had threatened to "bring her down" and then accused her of releasing his employment records in the custody case, only to have it turn out that he had signed an authorization for their release himself. There was also a confirmed incident of illegal moose-hunting (which is apparently a big deal up there) and allegations of drinking on the job.
Everyone who has had a violent armed nutcase like that in the family will understand the concerns. Palin's picture could end up hung in the place of honor in battered women's shelters around the country.
2. If this were a Biden-Obama ticket, criticizing her inexperience would carry little enough weight. Given an Obama-Biden ticket, the cricitism sounds like a parody.