We present a combined experimental-theoretical study on structural and coverage dependences of the adsorption and desorption of molecular hydrogen on atomically flat Cu(111) and highly stepped Cu(211) surfaces. For molecules with identical incident energy from supersonic molecular beams, we find a reduced dissociative sticking probability for the stepped surface compared to Cu(111). DFT calculations of activation barriers to dissociation for the clean and partially precovered surfaces, as well as quantitative analysis of TPD spectra, support that the A-type step of the (211) surface causes an upward shift in activation barriers to dissociation and lowering of the desorption barrier. The new data allow us to determine low sticking probabilities at conditions where King and Wells measurements fail to determine the reactivity. They are also fully consistent with the unexpected observation that monoatomic steps on a surface lower the reactivity toward the dissociation of a diatomic molecule.