In this work, we investigate the impact of noise on two-degree-of-freedom Hamiltonian systems where chaotic scattering takes place. We show that noise influences the system differently based on whether the isopotential curves are open or closed. Within the open regime, noise induces a phenomenon of noise-enhanced trapping, leading to an increase in the average escape time for particular noise intensities. On the other hand, in the closed regime noise can generate escapes that are not possible in the conservative system.