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).

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

Numerical criteria for base-pointfreeness are known only in specific cases such as the Kawamata basepoint-free theorem and Reider's theorem (for $\dim X=2$).

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 the 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).

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