In order to prevent early distress in asphalt pavement and save on subsequent operational and maintenance costs, modifying asphalt is an effective approach. Styrene–butadiene–styrene (SBS) block copolymers, due to their excellent physicochemical properties, have become a mature and widely used asphalt modifier. Carbon nanotubes (CNTs) possess advantages such as a large specific surface area and high modulus, which, when incorporated into asphalt, can enhance its deformation resistance. To analyze the effect of incorporating CNTs on SBS-modified asphalt (SBS-A), this study analyzed the influence of different CNT concentrations on the high and low-temperature performance and aging properties of SBS-A through penetration, softening point, ductility, dynamic shear rheometry, and short-term aging tests. The optimal CNT concentration was determined to be 1.0%. Furthermore, the changes in the modified asphalt during the aging process were analyzed using infrared spectroscopy.