Asteroid mining has been proposed as an approach to complement Earth-based supplies of rare earth metals and supplying resources in space, such as water. Existing research on asteroid mining has mainly looked into its economic viability, technological feasibility, cartography of asteroids, and legal aspects. More recently, potential environmental benefits for asteroid mining have been considered. However, no quantitative estimate of these benefits has been given. This paper attempts to determine if and under which conditions asteroid mining would have environmental benefits, compared to either Earth-based mining or launching equipment and resources into space. We focus on two cases: Water supply to cis-lunar orbit and platinum mining. First, we conduct a state-of-the-art of current environmental life cycle assessment for the space domain and platinum mining. Second, a first order environmental life cycle assessment is conducted, including goal and scope definition, inventory analysis, and impact assessment. We compare water supply to cis-lunar orbit with and without asteroid mining and go on to compare terrestrial with space-based platinum mining. The results indicate that asteroid water mining would have environmental benefits, as soon as the amount of water supplied via mining is larger than the mass of the spacecraft used for mining. For platinum mining, we find that by comparing the operations phase of terrestrial and space mining, space mining would have a lower environmental impact, if the spacecraft is able to return between 0.3 to 7% of its mass in platinum to Earth, assuming 100% primary platinum or 100% secondary platinum, respectively. For future work, we propose a more detailed analysis, based on a more precise inventory and a larger system boundary, including the production of the launcher and spacecraft.