Should a teacher help if the student does not ask for it? Alice is teaching a class and one student is clearly not understanding the material very well. Should she approach the student (e.g. ask to meet after class, write an email, or during class itself if it's not a lecture) and offer to help?

The argument for "yes" is that although the student might not ask for help, he or she wants to pass the course, so the teacher's intervention is appropriate. Some students also come from conservative cultures, are afraid to be seen as a fool, or are simply shy.

The argument for "no" is that it's possible the student wants to puzzle it out himself/herself. Instead, Alice should just make herself available for consultation. If the student does not make use of all the resources at his or her disposal, it's not her fault.

Is there any evidence that one style of teaching or the other is preferable?
Part of teaching is figuring out if what you want to teach is actually understood. This is something you should be doing continuously. As part of that you automatically find the students that struggle more than others, and that typically offers natural moments in which you can discuss what that student can do to improve things. So I don't see how someone can teach effectively without answering "yes" to your question.