In this paper, we propose a channel state adaptive random access for a wireless local area network (WLAN) where multiple receive antennas are used for space division multiple access (SDMA). In contrast to conventional medium access control (MAC) protocols which were designed separately from the characteristics of the physical (PHY) layer, our proposed protocol takes advantages of multi-packet reception (MPR) capability of the system which takes into consideration the channel state information (CSI). Specifically, each node's transmission is dynamically controlled based on the network population, current channel condition, as well as the number of receive antennas. The average frame error rate and throughput expression are derived as functions of the transmission control. An optimal transmission policy is then obtained to achieve the maximum throughput. Numerical results show that our cross-layer approach significantly outperforms conventional schemes where CSI is not taken into consideration or MPR is not allowed. The tradeoff between the performance improvement and hardware cost is also illustrated.