I don't think it's ever explicitly stated anywhere, but my gut feeling would be no. I would put her on a similar level as the other "support staff" type positions at Hogwarts, like Filch and Madam Pomfrey. Filch is never seen taking points from students, and I feel like he would take points constantly if it was something he could do. She (along with Filch and Pomfrey) doesn't seem high-ranking enough to take points. It's true that she is an authority figure, but I feel like if she could take points, we would have seen it at some point. For instance, the scene where she chases Harry out of the library for eating (fourth or fifth book?) would have been an ideal opportunity for her to dock a few points, but she simply shoos him away. I guess it's not impossible, but simply based on canon evidence, I would guess she doesn't have the privilege.
I looked through the forums and I couldn't find the thread of our lovely Hogwarts librarian. If it exists and I overlooked (which is very likely) please feel free to delete this thread.
My question is very simple. Is she able to take points from students? She never does it in the books, but it seems plausible to me since she's in charge of the library and has to make sure people behave, return books, among other things. On the other hand, she did have her fair share of opportunities to take points from Harry and she never did.