The adsorption of DNA or other polyelectrolyte molecules on charged membranes is a recurrent motif in soft matter and bionanotechnological systems. Two typical situations encountered are the deposition of single DNA chains onto substrates for further analysis, e.g., by force microscopy, or the pulling of polyelectrolytes into membrane nanopores, as in sequencing applications. In this paper, we present a theoretical analysis of such scenarios based on the self-consistent field theory approach, which allows us to address the important effect of charge correlations. We calculate the grand potential of a stiff polyelectrolyte immersed in an electrolyte in contact with a negatively charged dielectric membrane. For the sake of conciseness, we neglect conformational polymer fluctuations and model the molecule as a rigid charged line. At strongly charged membranes, the adsorbed counterions enhance the screening ability of the interfacial region. In the presence of highly charged polymers such as double-stranded DNA molecules close to the membrane, this enhanced interfacial screening dominates the mean-field level DNA-membrane repulsion and results in the adsorption of the DNA molecule to the surface. This picture provides a simple explanation for the recently observed DNA binding onto similarly charged substrates [G. L.-Caballero et al., Soft Matter 10, 2805 (2014)1744-683X10.1039/c3sm52428k] and points out charge correlations as a non-negligible ingredient of polymer-surface interactions.