(CNN) -- Caroline Kennedy is on a public campaign for Sen. Hillary Clinton's Senate seat, which may make it difficult for New York Gov. David Paterson not to send President John F. Kennedy's only living child to Washington.
Caroline Kennedy vists Syracuse, New York, on Wednesday.
While she has to win over only one voter -- Paterson -- Kennedy essentially began a political campaign for the seat, which was once held by her uncle, Robert F. Kennedy, with a visit to upstate New York on Wednesday.
And on Thursday, Kennedy met with civil rights activist Al Sharpton, an influence in national and New York politics, in Harlem.
"You know, I come at this as a mother, as a lawyer, as an author, an education advocate and ... from a family that really has spent generations in public service, and I feel this commitment," Kennedy said after her meeting with Sharpton. "This is a time when nobody can afford to sit this one out, and I hope that I have something to offer."
Kennedy added, "There are many qualified candidates in this race, and it is up to the governor. But this is a commitment I've made over many years, and I'd love to take the skills and relationships I have to Washington to fight for the people of New York."
New York law gives Paterson the sole power to decide who will succeed Clinton after she gives up her Senate seat to become President-elect Barack Obama's secretary of state.
Clinton has said she is staying out of the process of picking her successor and letting Paterson decide.
During a stop in Syracuse, New York, Kennedy said, "I've told Gov. Paterson that I'd be honored to be considered for the position of United States senator. I wanted to come upstate and meet with Mayor [Matthew] Driscoll and others to tell them about my experience and also to learn more about how Washington could help these communities."
But when pressed by reporters on what her qualifications for the office are, Kennedy initially provided little insight.
"There's a lot of good people ... that the governor is considering," she said, adding, "He's laid out a process, and I'm proud to be in that process."
Later, during a stop in Rochester, New York, Kennedy said, "I have had a lifetime commitment to public service. I've written book on the Constitution and the importance of individual participation." Watch Kennedy's trip to upstate New York »
But some are questioning whether she's ready to follow the path taken by her father and her two uncles. Kennedy, 51, has been involved in education reform and women's issues but has never held public office and has mostly stayed out of the public spotlight.
Kennedy's high-profile campaign -- and powerful backers like Senate Majority Leader Harry Reid -- may force Paterson's hand. Watch Reid back Kennedy »
"She's clearly floated to the top. If it's not a done deal, it's starting to look more and more like ... Gov. Paterson is boxed in," Democratic strategist Hillary Rosen said.
"By the way, it wouldn't be a bad choice," Rosen added. "She's a classy woman, deeply intelligent and compassionate. But from what I'm hearing from my sources, Gov. Paterson still wants that group of a few folks to stay in this, so he has a choice to make."
Recent polls suggest that Kennedy is not the top choice of the public, however. A poll released by Siena College on Wednesday found that 26 percent of New York voters wanted another Democrat with a famous last name to be appointed to the seat: state Attorney General Andrew Cuomo, son of former New York Gov. Mario Cuomo. Kennedy was the choice of 23 percent of New York voters, the polls found.
Sending Cuomo to the Senate, however, might anger feminist groups that are pressuring Paterson to appoint a woman to replace Clinton. Considering that Clinton's departure will leave only 16 women in the Senate out of 100 senators, many feminists believe women are underrepresented on Capitol Hill.
"It's really important to us that we don't lose another seat in the U.S. Senate," said Cathy Lasry, president of the Eleanor Roosevelt Legacy Committee. "I think that we've proven that once we are in there, obviously we can govern. ... We just need that extra push and some help in getting in there." Watch the Kennedys' hold on American politics »
Choosing Kennedy would allow Paterson to maintain the number of women in the Senate, but she is not the only female candidate the governor is reportedly considering. Rep. Carolyn Maloney, D-New York, has also expressed interest in the seat.
But one New York Republican believes that the number of women in the Senate shouldn't influence Paterson's pick and that Kennedy is not ready for the job.
"We should go beyond gender politics and racial politics, and we should pick the best person for the job," said Rep. Peter King, who was mentioned as a candidate for the Senate seat in 2010. "I quite frankly don't think Caroline Kennedy has shown the experience or the qualifications to be a United States senator." Watch King question Kennedy's qualifications »
Democratic strategist Hank Scheinkopf, who once worked for President Bill Clinton, also said he thought Kennedy should not be chosen.
"There are a lot of Democrats who aren't happy about this, frankly," Scheinkopf said. "Why go outside the home team when there's a lot of good players here?"
But Kennedy would continue one of America's most powerful political legacies, which will force Paterson to take a hard look.
"As a woman, as a feminist, I think it's time to put another woman in the seat," CNN contributor and Democratic strategist Donna Brazile said. "I think Caroline Kennedy will bring a great deal to the table."
When asked about Kennedy on ABC's "The View" on Thursday, New York Mayor Michael Bloomberg, who had praised her as a candidate, said he would not back any of the candidates publicly, saying, "I should stay out of the race personally. ... It's not my call."
He said Paterson should pick "somebody who is smart, who can get access, who will hire good advisers, who has character and judgment."
When asked if Kennedy fit that profile, Bloomberg said, "yes, but there are other people as well."
CNN's Mary Snow contributed to this report.
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