Motivated by various disordered propagation problems with competing channels, I study the representative problem of Anderson localization on an asymmetric two-leg ladder. The problem is solved by the Fokker-Planck approach, which is exact in the weak disorder limit. The localization radius of various one dimensional systems, such as a polaritons or other hybrid particles, can be investigated by this model. These applications correspond to parametrically different intra-chain hopping integrals and/or different disorder amplitudes on the two legs, situations in which it is non-trivial to predict what dominates the transport in the joint system. An extended Dorokhov-Mello-Pereya-Kumar (DMPK) equation is obtained and solved analytically. Two localization lengths are obtained as functions of the parameters of the model. We find that: 1) Near the resonance energy (where the dispersion curves of the two decoupled and disorder-free chains intersect) the "slow'' chain dominates the localization properties of the ladder. 2) Away from the resonance the "fast'' chain dominates the transmission probability.