We study the dynamics of a mobile impurity in a two-leg bosonic ladder. The impurity moves both along and across the legs and interacts with a bath of interacting bosonic particles present in the ladder. We use both analytical (Tomonaga-Luttinger liquid) and numerical [Density Matrix Renormalization Group (DMRG)] methods to compute the Green's function of the impurity. We find that for a small impurity-bath interaction, the symmetric mode of the impurity effectively couples only to the gapless mode of the bath while the antisymmetric mode of the impurity couples to both gapped and gapless modes of the bath. We compute the time dependence of the Green's function of the impurity, for impurity created in either the antisymmetric or symmetric mode with a given momentum. The latter case leads to a decay as a power law below a critical momentum and exponential above, while the former case exhibits both power-law and exponential decay depending on the transverse tunneling of the impurity. We compare the DMRG results with analytical results using the linked cluster expansion and find good agreement. In addition, we use DMRG to extract the lifetime of the quasiparticle, when the Green's function decays exponentially. We also treat the case of an infinite bath-impurity coupling for which both the symmetric and antisymmetric modes are systematically affected. For this case, the impurity Green's function in the symmetric mode decays as a power law at zero momentum. The corresponding exponent increases with increasing transverse tunneling of the impurity. We compare our results with other impurity problems for which the motion of either the impurity or the bath is limited to a single chain. Finally, we comment on the consequences of our findings for experiments with the ultracold gases.