  This paper deals with the problem of computing a generating set for the Cox
ring $R(X)$ of a smooth projective rational surface $X$ with nef anticanonical
class. In case $R(X)$ is finitely generated, we show that the degrees of its
generators are either classes of negative curves, elements of the Hilbert basis
of the nef cone or certain ample classes of anticanonical degree one, which
only appear when $X$ is a rational elliptic surface of Halphen index $m>2$.
Moreover, we partially characterize which elements of the Hilbert basis of the
nef cone are irredundant for generating $R(X)$. We apply this result to compute
explicit minimal generating sets for Cox rings of some rational elliptic
surfaces of Halphen index $>1$.
