Metal membranes play a vital role in hydrogen purification. Defect-free membranes can exhibit effectively infinite selectivity but must also provide high fluxes, resistance to poisoning, long operational lifetimes, and low cost. Alloying offers one route to improve on membranes based on pure metals such as palladium. We show how ab initio calculations and coarse-grained modeling can accurately predict hydrogen fluxes through binary alloy membranes as functions of alloy composition, temperature, and pressure. Our approach, which requires no experimental input apart from knowledge of bulk crystal structures, is demonstrated for palladium-copper alloys, which show nontrivial behavior due to the existence of face-centered cubic and body-centered cubic crystal structures and have the potential to resist sulfur poisoning. The accuracy of our approach is examined by a comparison with extensive experiments using thick foils at elevated temperatures. Our experiments also demonstrate the ability of these membranes to resist poisoning by hydrogen sulfide.