I am a second-year graduate student in applied mathematics looking to apply for fellowships for this upcoming year. It also happens to be the case that I am a Caucasian male. Would it be more advantageous for me to select "Prefer not to respond" in applications that ask for my race and gender (especially applications that give preference to women and minority students), or is that generally considered to do more harm than good?
Technically, I see nothing wrong with not reporting your race/gender. However, I suspect that if a fellowship gives preference to a specific group, putting "prefer not to respond" puts you outside of that group by default. Therefore, it would make no difference if you were a Caucasian or unidentified, you would still not gain the selection benefit for minorities. Hence the net benefit would be zero.
Please don't do this. Race/gender information has two purposes: 1) understanding the demographics of the applicant population, and 2) ensuring that strong candidates from underrepresented groups are not overlooked (i.e. some degree of affirmative action).
Most descriptions of 2) that I have heard essentially provide benefits to people who are underrepresented, or have in the past been discriminated against. Listing yourself either as a Caucasian male or as [decline to state] is likely to take you out of that population. [Caveat: I don't review fellowships myself, this is only based on what people have told me about admissions committees, etc.; for a given fellowship, the details of this may be publicly available.]
However, if a large fraction of Caucasian males decline to state their race/gender, the statistics that the NSF/whoever collects on their applicant pool will be biased. This will lead the NSF to have erroneous conclusions about whether their efforts to encourage diversity are working.
In essence, doing this is poisoning a well - corrupting a public source of data for negligible personal benefit.