Is it appropriate for employers recruiting students to ask professors for syllabi of their classes?
I don't see why it would be an issue.

I've been in a similar position in the past - recruiting at the university that I graduated from. On every trip, I made it a point to visit the department office to catch up with the department chair (who was a professor when I was there), the academic advisers, and the professors. I never asked for a syllabus for courses, but I did make it a point to talk about changes to course content. I felt that doing this would help me stay closer to the department and college, but also would make me better at recruiting and interviewing since I would have a better understanding of what the students were learning in the classroom.

I would encourage you to, somehow, stay in touch with changes to the classes that are core in the departments that you recruit from so you can be a better interviewer. I would also encourage you to share this information with HR and hiring managers so they know the capabilities. If there are new courses or new content, it could make the hiring process for interns or new graduates easier.