Average Bit Error Rate (BER) expression of free-space optical (FSO) communication links with Airy beam as signal carrier under weak atmospheric turbulence and on-off keying (OOK) modulation scheme is derived based on scintillation index of Airy beam and Exponentiated Weibull (EW) channel model. The average BER has been evaluated at different transverse scale factors and exponential decay factors of Airy beam and link distances. And comparison of the average BER of FSO links with Airy beam and Gaussian beam as signal carrier has been carried out. The simulation results show that the average BER of FSO links with Airy beam as carrier decreases with the increase of mean signal to noise ratio (SNR) and increases with the increase of transmission distance. When the transverse scale factor is about 1.5cm, a lower average BER can be obtained. And the smaller the exponential decay factor is, the lower the average BER is. Under the same atmospheric turbulence condition, the average BER of FSO links with Airy beam as carrier is obviously better than that of FSO links with Gaussian beam as carrier. The results of this research have some significance for the application of Airy beam in FSO system.