In this letter, we analyze the secrecy performance of a reconfigurable intelligent surface (RIS)-aided communication system with spatially random unmanned aerial vehicles (UAVs) acting as eavesdroppers. We consider the scenarios where the base station (BS) is equipped with single and multiple antennas. The signal-to-noise ratios (SNRs) of the legitimate user and the eavesdroppers are derived analytically and approximated through a computationally effective method. The ergodic secrecy capacity is approximated and derived in closed-form expressions. Simulation results validate the accuracy of the analytical and approximate expressions and show the security-enhanced effect of the deployment of the RIS.