The existing fusion-based RGB-D salient object detection methods usually adopt the bistream structure to strike a balance in the fusion trade-off between RGB and depth (D). While the D quality usually varies among the scenes, the state-of-the-art bistream approaches are depth-quality-unaware, resulting in substantial difficulties in achieving complementary fusion status between RGB and D and leading to poor fusion results for low-quality D. Thus, this paper attempts to integrate a novel depth-quality-aware subnet into the classic bistream structure in order to assess the depth quality prior to conducting the selective RGB-D fusion. Compared to the SOTA bistream methods, the major advantage of our method is its ability to lessen the importance of the low-quality, no-contribution, or even negative-contribution D regions during RGB-D fusion, achieving a much improved complementary status between RGB and D. Our source code and data are available online at https://github.com/qdu1995/DQSD.