"I recognize that some members of the court have a different view, and certainly when and if I get to the court I will talk with them about that questions, but I have said that I think it would be a terrific thing to have cameras in the courtroom," Kagan said at C-SPAN cameras rolled (see above). "And the reason I think, is when you see what happens there, it's an inspiring sight. ... I basically attend every Supreme Court argument. ... It's an incredible sight, because all nine justices, they're so prepared, they're so smart, they're so thorough, they're so engaged, the questioning is rapid fire. You're really seeing an institution of government at work really in an admirable way." (Emphasis mine)
On the issue, the Justices range from willing to look at it to dead-set against it. The most common reason given for opposing them is a belief they would wreck the "mood" and "dynamic" of the Court.
While allowing cameras into the Court would certainly alter its dynamics, their inclusion would not irreparably harm the institution. I'm sure our representatives also weren't thrilled about allowing cameras into Congress, yet their value is obvious now.
By the time the Justices hear oral arguments, they and their cadre of clerks have already spent months reviewing case briefs and writing opinions and much of the legal legwork has already been done. The arguments simply allow the Justices to ask any questions that may remain.
Given this nation's lack of civic knowledge, maybe allowing cameras into the Court would go a long way in helping us better understand how the Judiciary impacts us all.
Here is some video containing Kagan's answer: