We establish large deviation principles and phase transition results for both quenched and annealed settings of nearest-neighbor random walks with constant drift in random nonnegative potentials on View the MathML source. We complement the analysis of M.P.W. Zerner [Directional decay of the Green’s function for a random nonnegative potential on View the MathML source, Ann. Appl. Probab. 8 (1996) 246–280], where a shape theorem on the Lyapunov functions and a large deviation principle in absence of the drift are achieved for the quenched setting.