In fact, it's African-Americans who are alleged to have done so:
“As long as the voter requests my help, I come in and assist them,” Mr. Turner said.
Blacks hold a majority on the five-member Perry County Commission. And an influential political organization, the Perry County Civic League, founded during the civil rights movement by one of the movement's leaders, Albert Turner Sr., plays a large role in filling other offices.
The Democracy Defense League, a biracial citizens group that has been critical of voting abuses in the area for the last three years, submitted affidavits to Mr. King and federal officials saying that some people voted six times in the same election.
The accusations have roiled this old plantation country of rich soil, impoverished people and unremitting distrust between black majorities and white minorities, where several trials and indictments in vote fraud cases have already occurred over the years—in 1985, 1997 and 2007—some of which resulted in convictions of county and city officials.
The accusations have larger national implications at a moment when many Republican officials around the country have cited rampant voter fraud as a justification for stricter identification requirements at the polls, a move that Democrats have said is really designed to dampen the turnout among Democratic voters. Specific examples of fraud, though, are quite unusual.
Some Democrats here say the inquiries are motivated by racism and partisanship.
“The Republican Party has an unscripted mandate to target Democratic counties, and African-Americans particularly,” said Albert Turner Jr., a county commissioner here.
Mr. Turner’s father, a local civil rights hero, was tried and acquitted on vote fraud charges 23 years ago. The younger Mr. Turner was photographed talking to voters in a polling place in the National Guard Armory throughout the day on June 3, when he was a candidate for re-election.
Although state law prohibits candidates from being within 30 feet of a polling place, Mr. Turner said another law allowed voters to choose someone to assist them in voting. He said he had helped as many as 60 people on primary day, when he was easily re-elected for a second term.
Officials Investigate 3 Alabama Counties in Voter Fraud Accusations, By Adam Nossiter, New York Times, July 10, 2008
"Assisting someone in voting" usually means going into the voting booth with someone who says they're illiterate or handicapped, and "helping" them vote. In old style machine politics, it was a way of making sure that your people voted the right way.D. W. Brogan wrote in Politics in America (1955) that
"One last resort of the machine is to take advantage of the law that permits an illiterate or physically incapacitated voter to be “assisted” in casting his vote. This is an effective method of controlling voters, especially in years like 1928 and 1932 in Philadelphia, when it was suspected that even the faithful could not be trusted alone behind the green curtain. To profess to be able to vote unaided was, in itself, ground for suspicion of intended treason and was effectually frowned on. As the number of foreign-born and of illiterates steadily drops, this device, though not impracticable, becomes more and more implausible."
Of course, when he wrote of the number of foreign-born and illiterates dropping, that was before the 1965 immigration act.
As for the candidate himself giving voter assistance, a civil rights magazine wrote, in regard to the 1985 investigations mentioned in the New York Times, that
"Under Alabama law, it is illegal (indeed, it is so written on the ballot) for candidates to witness absentee ballots or give assistance to voters."[Crackdown in the Black Belt: On to Greene County By Randall Williams, Southern Changes. Volume 7, Number 3-4, 1985]
It's obvious that if the candidate is the person who assists you, that he knows how you voted. They were accusing a Republican of having assisted voters, which is why they made point of it, but the law is the law.