Kagan is mostly criticized for her decision as Harvard Law Dean to relocate Military recruiters away from the school's career center because she opposed the military's "don't ask, don't tell" policy.
There are also fears that Kagan will allow her political views to influence her rulings. As a Supreme Court judge, it's important to stay completely neutral and make your ruling based on interpretation of the law. It's not the job of the Supreme Court to make law - it's their job to enforce it.
There are also fears that Kagan doesn't believe in the fundamental right to bear arms. Considering that recent Supreme Court ruling which deemed the long-time Chicago ban on handguns unconstitutional has been made, many can't help but wonder how that ruling would have differed had Kagan been on the bench.
Kagan also has been viewed as not ready for such a post, as she doesn't have any bench experience. She does have extensive experience as an attorney and as the Dean of Harvard Law.
There are few people who doubt that Kagan will eventually be confirmed. However, just as with a lot of other measures that require Senate approval Kagan's confirmation will be met with at least some opposition, and that's only to be expected.