In this paper, the multicode (mc) and the variable spreading gain (VSG) access schemes for asynchronous space-time block coded (STBC) multicarrier code division multiple access (MC-CDMA) systems over frequency selective Rayleigh fading channels are considered. The system capacities for these two multirate access schemes are investigated. Transmit power control is proposed to maintain the link quality and improve the system capacity. Numerical results show that the system capacities for these two access schemes, when orthogonal Gold spreading sequences are used, have nearly the same system capacity. In the case when non-orthogonal Gold codes are used, the mc access scheme shows a degradation in the system capacity as compared to VSG, due to the presence of larger self-interference (SI) among the sequences used by each high rate user.