In The Geometry of Schemes by Eisenbud and Harris, Exercise I32 asks one to show that a scheme $X$ is reduced if and only if every local ring $\mathcal{O}_{X,p}$ is reduced for closed points $p \in X$. However, this does not seem to work in general, since $X$ may not have enough closed points. What additional hypotheses on $X$ do I need for such an assertion to hold?

There do exist schemes without a closed point, yes. (Liu, exercises 3.3.26/27) But under some very reasonable additional conditions  I think quasicompactness will be sufficient, if you are happy with using Zorn's lemma  the result holds. Use/prove the existence of a closed point, and the fact that localizing a reduced ring still gives you a reduced ring. 


It seems to me that looking at closed points only is not sufficient since they are not always a dense set of X ... 


I don't think quasicompactness is enough,for Noether scheme it is true. in a noether scheme, every point P has a closed point in its closure, so ..... but i don't find a necessary and sufficient condition 

