Actually, I'm not even sure if "sexist" is the right word here. The predominant criticism of River Song isn't so much that she's sexist, although you do get a good few people in there engaging in the popular pastime of "faux-feminist slut-shaming"; it's that she's nothing but a gigantic Mary-Sue that Moffat has inserted into the story. According to this contingent, Moffat is making all the plots revolve around River Song, to the point where she's better than the Doctor at everything and all the important plot points involve her.
To some extent, this is simply a case of not knowing what words mean. A "Mary Sue" is a somewhat vague term, but is generally taken to be an overly idealized, absurdly talented, one-dimensional author surrogate, who is improbably lucky and succeeds where the main character fails, and who is the center of the story even in a canon that has not previously included her. That's the basic definition, as crowdsourced by the Internet.
Now let's look at River song. Since her first appearance, she's been in nine episodes. Out of 37. (Technically ten, if you want to count her cameo at the end of "Closing Time".) If this is a series that "revolves around" her, it's in a very eccentric orbit to say the least. Yes, she features prominently in the Silence arc. That's because the Silence's plan was to use her to kill the Doctor. She failed at that pretty much completely...which pretty much takes care of "improbably lucky and succeeds where the main character fails", too. The only thing she really succeeded at was at getting electrocuted better than the Doctor.
Is she more talented than the Doctor? Many people point to the fact that she can fly the TARDIS better than he can...which would be a totally excellent argument to make if it wasn't a 49-year-long running gag that for all the Doctor's Time Lord training and experience, he's actually a lousy pilot who can't get the TARDIS where he wants it to go better than one in three tries. It's the same joke they did when Romana joined the cast, and that was quite literally before lots of current fans were born.
Overly idealized, one-dimensional author surrogate? Um...unless there's something fairly important about Steven Moffat we don't know, I don't think that he really wants to be a sexy woman who shoots well, rides horses and has hallucinogenic lipstick. He wrote her as a love interest for the Doctor, yes, and he assumed that any love interest for the Doctor has to be something more than a simpering damsel in distress for him to rescue...which is a GOOD THING...but he definitely worked hard to make her more than one-dimensional. Her backstory is interesting, complicated, and paints her as more than a little damaged. Idealized, she ain't.
Which ultimately leaves us with, "She seems to know more than the Doctor. She seems to be as good at things as him. She's talented, and that makes her threatening." Which is, when you break it down, kind of a sexist argument for disliking a character. Not that this is anything new; pretty much any talented female character in fiction is going to get tagged with "Mary Sue" at some point, because women who are good at things and are admired within the story for it are always Mary Sues. (Somehow, Captain Kirk is never considered to be a Mary Sue.) Some people will always seek to tear down any female character who seems skilled and competent, because they view this as exclusively the purview of male characters. They might dress that up as concern, but it's fundamentally a sexist point of view.
Oh, plus she is naked in some scenes and thus has lady parts that she displays without shame, which makes her a Bad Person. (See previous essay.)
So is River Song sexist? Weeeeeeelll....maybe a little. Because while the Doctor's story definitely doesn't revolve around her, it's become increasingly clear that her story revolves around the Doctor. She didn't become an archaeologist because she was interested in old things, as we originally thought; she became an archaeologist to track down the Doctor. She loves the Doctor, she goes to jail for the Doctor, she occasionally attempts to kill the Doctor...there's very little in River Song's story that she does simply because she wants to. It's hard not to argue that this is a somewhat sexist angle on her character, and that she was more interesting before she became the Doctor's "bespoke assassin". Does that make her "sexist"? Probably not by itself. But it's certainly problematic.
But unfortunately, as with Amy Pond, the most sexist thing about River Song is the ways that people find to complain about her.