We investigate adsorption and passive translocation of random amphiphilic copolymers interacting with a self-assembled lipid bilayer membrane. By using the bond fluctuation model with explicit solvent, we consider random copolymers under variation of the fraction, H of hydrophobic sites and chain length. Our results indicate a point of balanced hydrophobicity, where a slight excess of hydrophobic monomers compensates an additional insertion barrier due to the self-organized packing of the bilayer. Close to balanced hydrophobicity, we observe translocation events of shorter polymers through the membrane. Compared to homopolymers, surface localization of amphiphilic polymers is considerably increased due to the polar nature of the molecules with respect to the amphiphilic environment, and translocations are suppressed for longer chains. Close to balanced hydrophobicity, the polymer induces dynamic and static perturbations in the bilayer, and permeability with respect to solvent is significantly increased around the copolymer. We discuss how to design membrane-active copolymers with a desired emphasis on either translocation or permeabilization based on a systematic sequence analysis. Our results indicate that alternating copolymers with an optimal block size smaller than the lipid size maximize perturbation of the bilayer, whereas for passive translocation, the limit of small block size or homopolymers with balanced hydrophobicity are most relevant.