In order to investigate the influence of basalt fibers (BFs) on the mechanical performance of recycled aggregate concrete (RAC), some groups of RAC specimens were first tested involving different types of fibers such as carbon fibers, steel fibers, polypropylene fibers and hybrid fibers. The main four indices for the investigation consisted of cube compressive strengths, axial compressive strengths, splitting tensile strengths and Young’s modulus. The effects of fiber volume fractions on the RAC slumps were also discussed. Meanwhile, the mechanical properties and failure modes of the BF-reinforced RAC were compared with those of other fiber-reinforced RAC and common concrete (CC). Subsequently the optimal volume fractions of BFs were explored for different mechanical properties within the volume fraction range of 0–0.2%. The back propagation neural networks were further applied to predict and validate the optimal BF fractions. Lastly, the general strength formulas, as well as the elastic modulus formula, for BF-reinforced RAC were deducted based on the specimen test results. It is found that the addition of fibers may improve the failure modes of RAC and different fibers present positive or negative effects on the mechanical properties. The optimal volume fractions of BF with respect to the four mechanical indices are 0.1%, 0.15%, 0.1% and 0.2% respectively. The proposed strength and elastic modulus formulas of BF-reinforced RAC provide satisfactory predictions with the test results and thus can be used as a reference in practice.