No general numerical criterion for base-pointfreeness is known except in specific cases such as the Kawamata basepoint-free theorem and Reider's theorem for surfaces. 

In the case you mention, $X=\mathcal{M}_{0,n}$, the problem of classifying semi-ample divisors is an important problem.  It is slightly easier in positive characteristic, thanks to a theorem of Keel which says that a nef line bundle $L$ is semi-ample if and only if the restriction $L|_E$ is semiample, where $E$ is the exceptional locus of subvarieties $Z$ such that $L^{\dim Z}.Z=0$. If $f:X\to Y$ is a morphism with exceptional locus $E$, then $L$ is semi-ample if and only $L^r$ is the pullback of an ample line bundle on $Y$ for $r>0$. For more precise statements, you might want to take a look at

[S. Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Annals of Mathematics(1999)][1].

According to G. Farkas' [article][2], there are currently no known examples of nef divisors which are not semi-ample.


  [1]: http://www.kurims.kyoto-u.ac.jp/EMIS/journals/Annals/149_1/keel.pdf
  [2]: http://www-irm.mathematik.hu-berlin.de/~farkas/seattle.pdf