Clathrin-coated structures must assemble on cell membranes to perform their primary function of receptor internalization. These structures show marked plasticity and instability, but what conditions are necessary to stabilize against disassembly have not been quantified. Recent in vitro fluorescence experiments have measured kinetics of stable clathrin assembly on membranes as controlled by key adaptor proteins like AP-2. Here, we combine this experimental data with microscopic reaction-diffusion simulations and theory to quantify mechanisms of stable vs unstable clathrin assembly on membranes. Both adaptor binding and dimensional reduction on the 2D surface are necessary to reproduce the cooperative kinetics of assembly. By applying our model to more physiologic-like conditions, where the stoichiometry and volume to area ratio are significantly lower than in vitro, we show that the critical nucleus contains ~25 clathrin, remarkably similar to sizes of abortive structures observed in vivo. Stable nucleation requires a stoichiometry of adaptor to clathrin that exceeds 1:1, meaning that AP-2 on its own has too few copies to nucleate lattices. Increasing adaptor concentration increases lattice sizes and nucleation speeds. For curved clathrin cages, we quantify both the cost of bending the membrane and the stabilization required to nucleate cages in solution. We find the energetics are comparable, suggesting that curving the lattice could offset the bending energy cost. Our model predicts how adaptor density controls stabilization of clathrin-coated structures against spontaneous disassembly, and shows remodeling and disassembly does not require ATPases, which is a critical advance towards predicting control of productive vesicle formation.