How to handle a student's public speaking anxiety when teaching a class that requires presentations?
Fear of speaking in public is common. It is one of the major symptoms of social phobia. I'll work on the assumption that your student has diagnosed or undiagnosed SP (I used to work on SP data.)

SP is very amenable to classical cognitive behavioral therapy (CBT). You are a teacher, not a therapist. I would counsel against "rolling your own" therapy program. Instead, refer the student to on-campus counseling and strongly encourage her to seek professional help. As others write, the need to present will not go away; neither at college nor at the workplace. And college is probably the best place to start therapy. To repeat: this is a good point in time to start working on this; encourage your student to do so with a therapist.



So, that's the long-term answer I'd give. However, even if CBT frequently shows effects rather quickly in SP, your student will likely not have progressed far enough in her therapy to actually present her paper by the time she should. So you will likely need to make some kind of accommodation. I'd recommend that you contact your university's student service. There likely are some sort of guidelines on how to deal with such an issue - you probably are not the first professor to be confronted with this.

If you are afraid of getting a reputation as a professor who can be swayed, there are different ways around this:


You could require an official diagnosis of SP, or whatever doctor's communication your local privacy laws permit. In this, you could let yourself be guided by what kind of paperwork is required for a student to skip and retake an exam for medical reasons. (Student services could again be helpful here.)
You could make sure that the "alternative assignment" is not a bonus, but does add significant work to the student's workload. I personally would be careful here, since you don't want to penalize her for her disability.




That said, if the student does take your recommendation and starts therapy, it is quite possible that her therapist would assign making at least some kind of presentation as a piece of "homework" for her. Apart from possible confidentiality issues (your student may opt for waiving these, so you could talk directly to her therapist), this could allow you to help her much more than just doing a short-term fix. It may be worthwhile pointing this out to her. After all, you are willing to go to an effort to help her - her next professor or her employer may be less accommodating.