The structural properties and the energetics of clusters of argon and neon atoms containing the atomic impurity H-, NenH-, and ArnH-, with n varying from 9 up to 25, are examined. The same calculations are also carried out for the corresponding neutral homogeneous clusters, Ne-n and Ar-n. The results of the calculations, the physical reliability of the present modeling of the interactions, and the similarities, as well as the differences, between the anionic and the neutral complexes are discussed in some detail. The emerging picture shows that the dopant atom H- always locates itself outside the Ne-n moiety without significantly affecting the overall geometry of the cluster, while on the other hand, the ArnH- and Arn+1 clusters present very similar structures in which the H- dopant replaces one of the inner Ar atoms and therefore becomes fully "solvated" within the cluster.