Nucleic acids and lipids function in close proximity in biological processes, as well as in nanoengineered constructs for therapeutic applications. As both molecules carry a rich charge profile, and frequently coexist in complex ionic solutions, the electrostatics surely play a pivotal role in interactions between them. Here we discuss how each component of a DNA/ion/lipid system determines its electrostatic attachment. We examine membrane binding of a library of DNA molecules varying from nanoengineered DNA origami through plasmids to short DNA domains, demonstrating the interplay between the molecular structure of the nucleic acid and the phase of lipid bilayers. Furthermore, the magnitude of DNA/lipid interactions is tuned by varying the concentration of magnesium ions in the physiologically relevant range. Notably, we observe that the structural and mechanical properties of DNA are critical in determining its attachment to lipid bilayers and demonstrate that binding is correlated positively with the size, and negatively with the flexibility of the nucleic acid. The findings are utilized in a proof-of-concept comparison of membrane interactions of two DNA origami designs - potential nanotherapeutic platforms - showing how the results can have a direct impact on the choice of DNA geometry for biotechnological applications.