Context. So far, more than 130 extrasolar planets have been found in multiple stellar systems. Dynamical simulations show that the outcome of the planetary formation process can lead to different planetary architecture (i.e. location, size, mass, and water content) when the star system is single or double. Aims. In the late phase of planetary formation, when embryo-sized objects dominate the inner region of the system, asteroids are also present and can provide additional material for objects inside the habitable zone (HZ). In this study, we make a comparison of several binary star systems and how efficient they are at moving icy asteroids from beyond the snow line into orbits crossing the HZ. Methods. We modelled a belt of 10 000 asteroids (remnants from the late phase of the planetary formation process) beyond the snow line. The planetesimals are placed randomly around the primary star and move under the gravitational influence of the two stars and a gas giant. As the planetesimals do not interact with each other, we divided the belt into 100 subrings which were integrated separately. In this statistical study, several double star configurations with a G-type star as primary are investigated. Results. Our results show that small bodies also participate in bearing a non-negligible amount of water to the HZ. The proximity of a companion moving on an eccentric orbit increases the flux of asteroids to the HZ, which could result in a more efficient water transport on a short timescale, causing a heavy bombardment. In contrast to asteroids moving under the gravitational perturbations of one G-type star and a gas giant, we show that the presence of a companion star not only favours a faster depletion of our disk of planetesimals, but can also bring 4-5 times more water into the whole HZ.