Rigid DNA nanostructures that bind to floppy bilayer membranes are of fundamental interest as they replicate biological cytoskeletons for synthetic biology, biosensing, and biological research. Here, we establish principles underpinning the controlled interaction of DNA structures and lipid bilayers. As membrane anchors mediate interaction, more than 20 versions of a core DNA nanostructure are built each carrying up to five individual cholesterol anchors of different steric accessibility within the 3D geometry. The structures binding to membrane vesicles of tunable curvature is determined with ensemble methods and by single-molecule localization microscopy. This screen yields quantitative and unexpected insight on which steric anchor points cause efficient binding. Strikingly, defined nanostructures with a single molecular anchor discriminate effectively between vesicles of different nanoscale curvatures which may be exploited to discern diagnostically relevant membrane vesicles based on size. Furthermore, we reveal anchor-mediated bilayer interaction to be co-controlled by non-lipidated DNA regions and localized membrane curvatures stemming from heterogenous lipid composition, which modifies existing biophysical models. Our study extends DNA nanotechnology to control interactions with bilayer membranes and thereby facilitate the design of nanodevices for vesicle-based diagnostics, biosensing, and protocells.