On March 25, 2008, at Butler University in Indiana, Chelsea Clinton was asked whether her mother's credibility had been harmed as a result of her defense of the president. (Hillary Clinton had defended her husband during the scandal, though many of the allegations made against President Clinton -- in particular, that he had an affair with Lewinsky -- later proved to be true.)
Chelsea Clinton responded by saying:
You're the first person, actually, to ever ask me that question in the -- I don't know -- maybe 70 college campuses that I've now been to. And I do not think that's any of your business.
A week later, on April 1, at N.C. State in North Carolina, Chelsea Clinton was asked by a student why she insisted that the matter was not anyone's business. She replied:
It's none of your business. ... Well sir, I respectfully disagree, I think that is something that is personal to my family and I'm sure there are things that are personal to your family that you don't think are anyone's business either.
Chelsea Clinton's argument, then, is that the Monica Lewinsky scandal is a personal matter for her and her family. As a matter of privacy, it is not something that should be discussed. Or, at least, it is not something that Chelsea Clinton (or, perhaps, her family) should have to discuss.
However, this argument is based on a flawed premise. The Monica Lewinsky scandal was not -- at least, not entirely -- a personal matter for the Clinton family. Central to the scandal were accusations of perjury, obstruction of justice, and other allegations of attempts by President Clinton to thwart the judicial system. The consideration of such charges is a matter of public interest (even if the suspect is ultimately acquitted of the charges, as President Clinton was).
In addition, beyond the judicial aspects of the scandal, there is a genuine debate as to whether President Clinton's behavior with Monica Lewinsky demonstrated a critical lack of character. Not everyone believes that character is a legitimate political issue (and some who do don't believe that President Clinton’s character was significantly flawed), but the matter of character in politics is not straightforwardly spurious, either.
Certainly, some aspects of the Lewinsky scandal are personal -- and undoubtedly painful -- to the Clinton family. They should not have to answer questions about those aspects, and it is arguably inappropriate for anyone to ask such questions in the first place.
But there are plenty of questions regarding the Lewinsky scandal that are entirely legitimate matters of public concern. It is wrong for Chelsea Clinton to dismiss any and all questions on this topic as being inappropriately personal. Her appeal to privacy is unfounded.
(Aside on MSNBC source: In the MSNBC video, anchor Brian Williams says that Chelsea Clinton has been taking questions "always from the public, never from the news media, presumably to avoid tough questions about things like family history". Williams doesn’t defend this presumption. Is there evidence that the media tend to ask tougher questions than the public does, on this or any other issue?)