You should have someone who he looks up to talk to him about how excuses don't win races. You should also get him to realize that worring over a race isn't going to improve his situation. Instead, he should be getting fired up about races, thinking of how well he can do. Is he at the back of your varsity team or does he have some people around his speed level? If he has someone he can race with, I would highly recommend having them work together.
The one year I focused on bringing my high school team together, we had 5 runners finish within 5 seconds of one another during various races (16:10-16:20ish), but it involved a lot of yelling and encouraging during training and even in races. My team finished 2nd in the California State Meet for Division 3. If you have a "pack" of runners that he can generally keep up with, I would recommend trying to encourage him along during practice and races if your aerobic fitness allows you to do so (which is not an easy feat).
If he is that integral to your success as a team, then your team should take it as their responsibility to keep him afloat. Also, the more running you do in a pack like what I described above, the closer those team mates become, and the more of an impact your words will have on him (for allowing him to reframe his nervousness).
