Dynamical simulations show that the outcome of planetary formation process can lead to various planetary architectures (i.e. location, size, mass and water content) when the star system is single or double. In the late phase of planetary formation, when embryosized 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' characteristics and their efficiency to move icy asteroids from beyond the snow-line into orbits crossing the HZ. In our results, we highlight the key role of secular and mean motion resonances, causing an efficient flux of asteroids to the HZ on a short timescale. This in turn leads to asteroids bearing a non negligeable amount of water towards the HZ and available for any planets or embryos moving in this area. We also discuss how mass loss mechanisms can alter the water content on asteroids' surface.