The incorporation of basalt fiber into asphalt mixtures offers potential improvements in their viscoelastic properties. This study explores the addition of basalt fiber to Styrene Butadiene Styrene (SBS)-modified asphalt mixtures with varying SBS contents. Specifically, 0.3% basalt fiber was added to an asphalt mixture containing 3% SBS, and its performance, measured in terms of dynamic stability and flexural strength, was compared with a mixture with 7% SBS content. Additionally, finite element analysis using the Modified Burger’s Logit model was conducted to assess rutting and fatigue behavior. Given the high cost associated with increasing the SBS content, basalt fiber presents a cost-effective alternative without sacrificing performance. Laboratory tests, including the Marshall stability test, dynamic stability, flexural strength, and fatigue tests, were conducted to evaluate both mixtures. Results indicate that the mixture with 0.3% basalt fiber and 3% SBS outperforms the 7% SBS mixture, showing a 47% improvement in dynamic stability and rutting resistance and a 16% increase in flexural strength.