As a commonly used surface structure for airport runways, concrete slabs are subjected to various complex and random loads for a long time, and it is necessary to investigate their fracture performance at different strain rates. In this study, three-point bending fracture tests were conducted using ordinary performance concrete (OPC) and basalt fiber-reinforced airport pavement concrete (BFAPC) with fiber volume contents of 0.2, 0.4, and 0.6%, at five strain rates (10−6 s−1, 10−5 s−1, 10−4 s−1, 10−3 s−1, and 10−2 s−1). Considering parameters such as the peak load, initial cracking load, double K fracture toughness, fracture energy, and critical crack expansion rate, the effects of the fiber volume content and strain rate on the fracture performance of concrete were systematically studied. The results indicate that these fracture parameters of OPC and BFAPC have an obvious strain rate dependence; in particular, the strain rate has a positive linear relationship with peak load and fracture energy, and a positive exponential relationship with the critical crack growth rate. Compared with OPC, the addition of basalt fiber (BF) can improve the fracture performance of airport pavement concrete, to a certain extent, where 0.4% and 0.6% fiber content were the most effective in enhancing the fracture properties of concrete under strain rates of 10−6–10−5 s−1 and 10−4–10−2 s−1, respectively. From the point of view of the critical crack growth rate, it is shown that the addition of BF can inhibit the crack growth of concrete. In this study, the fracture properties of BFAPC were evaluated at different strain rates, providing an important basis for the application of BFAPC in airport pavement.