At the same time, it was suggested that Republicans might want her to be chosen, preferring to run against her in two years' time than against a more experienced politician.
Speaking on CBS' Face The Nation, Rep. Peter King, R-N.Y. (who has said he wants to run for Senate) called Kennedy a "People Magazine celebrity" who, as far as he knew, "has never held a real job.
"How can the average New Yorker identify with Caroline Kennedy?" King asked. "She comes from an outstanding family, I'm sure she's a wonderful parent, but she's never taken a stand on any public issue. She's never even held one news conference. She hasn't gone to one American Legion hall or Knights of Columbus hall or Masonic temple, or one synagogue to answer questions. When she does go on her so-called listening tour upstate, she's, like, running from city hall into the car to avoid reporters.
"And we just can't afford that. We can't afford to have a senator who is not prepared on day one."
While King said that other potential replacements for Hillary Clinton had "far more credentials" than Kennedy, he believed that choosing her would put Kennedy under tremendous scrutiny. "There's going to be a magnifying glass on her," he said. "And people are going to see that she is not equipped for the job. And I think the Republican candidate in 2010 will have a much better opportunity against Caroline Kennedy than against the others, especially someone who can identify with middle-income, working-class families."
The GOP sentiments were echoed by Rep. Tom Reynolds, from Buffalo in upstate New York, who decried the Senate as becoming "a House of Lords."
Referring to replacements being named or proferred to replace Senators Joe Biden and Barack Obama, Reynolds said, "We're seeing a seat-warmer in Delaware, a seat-seller in Illinois, and we're making seat-cushions in New York for, kind of, an aristocrat royalty of entitlement coming in here."
Those characterizations of Kennedy as an elitist were not accepted by New York City Schools Chancellor Joel Klein, who defended her experience, particularly in public service.
"I think people are missing the essence of this woman," Klein told host Bob Schieffer. "And I'll give you two instances. Five and a half, six years ago, when I asked her, along with Mayor Bloomberg, to come help with New York City's public schools, that's not a glamorous thing to do. This woman put her shoulder to the wheel, went out there and sent a powerful message to the entire city that public education matters. She helped us raise money. She helped us forge partnerships. And she spent time with our kids in the schools. I think that says a lot about her.
"My good friend Pete King said, you know, what does she know about the middle class or the working class, the struggling people? With all due respect, she came to work for the kids in New York. Most of those kids grow up in poor families."
Saying he has known Kennedy for a long time, Klein described her as "a very bright, very committed woman," and said that the work she did for President-elect Obama as part of his vice presidential search committee led her to become more involved publicly in issues of importance to her.
"I think in her own quiet but very effective way, she's going to be able to bring new ideas at a time when this country needs them," he said. "I think Caroline Kennedy has all of the right stuff. I think she's ready."
Fellow Democratic Rep. Gary Ackerman of New York, however, was less receptive to a return to Camelot:
"Everyone knows who she is, but we're not sure what she is," he said. "You know, they're Kennedys. They're all boats, but is she a sailboat when we need a battleship?"
He accused her handlers of protecting her (a la Sarah Palin's campaign staff) from reporters. "They're kind of building a mystique and an industry around her, when we need somebody to fight.
"She has a very famous name and she's a very attractive candidate. Those are good things in politics. But it's not an entitlement. You know, one of the things that we have to observe is that DNA, in this business, can take you just so far. You know, Rembrandt was a great artist. His brother Murray, on the other hand, Murray Rembrandt wouldn't paint a house."
Also on the program, former Rep. Geraldine Ferraro, D-N.Y., said she recommended to Gov. David Paterson that whomever he selects to replace Clinton should be someone who is already familiar with Congress in order to effectively deal with the myriad problems being faced in Washington and Albany.
"In order to get somebody to hit the ground running, I specifically said [to the governor] I think you should look to members of Congress," she said. "They've been dealing with these issues for the last several months. … They know what's going on, they know the process.
"And then I did put it in a little bit of a support for the six women members. I said I don't know if any one of them wants it, but I think you should reach out to them and ask them. That was it."
At the conclusion of the program, political anchor Dominic Carter of the New York City cable news channel NY1 said that, from what he has heard, "It is basically a done deal - that if Caroline Kennedy wants this seat (and she has expressed interest in it), then it is hers."
He disputed remarks that Kennedy was unprepared or unwilling to face the media. "Last week I had the opportunity to sit down with her for a couple of minutes up in Harlem, and the Camelot signs are there," he told Schieffer. "When she walked into this restaurant in Harlem, the crowd gave her basically a standing ovation. And then she sat down for a private event. I saw confidence. I saw that she wasn't looking to run away from the media. It has been somewhat of a slow rollout of Caroline Kennedy, and I think her folks are taking the heat for that."
Carter also said that the backing of Obama and Senate Majority Leader Harry Reid will help convince Gov. Paterson to choose her, especially looking ahead to 2010, when both her seat and that of fellow New Yorker Chuck Schumer will be up for reelection.
"She brings prestige, she brings the Kennedy name, and she brings the ability to raise a lot of money," Carter said. "That's one less problem the Democrats would have to worry about."
And what about Hillary? Would she support Kennedy, after Kennedy backed Obama?
"She has told her people to back off," Carter said. "She is moving ahead in terms of becoming Secretary of State. And you know, for all the criticism we've heard of Caroline Kennedy, we've been there before in terms of hearing the exact same thing about Hillary Clinton."
Read the full "Face the Nation" transcript here.
By CBSNews.com producer David Morgan.