In the paper, we investigate the coordination process of sensing and computation offloading in a reconfigurable intelligent surface (RIS)-aided base station (BS)-centric symbiotic radio (SR) systems. Specifically, the Internet-of-Things (IoT) devices first sense data from environment and then tackle the data locally or offload the data to BS for remote computing, while RISs are leveraged to enhance the quality of blocked channels and also act as IoT devices to transmit its sensed data. To explore the mechanism of cooperative sensing and computation offloading in this system, we aim at maximizing the total completed sensed bits of all users and RISs by jointly optimizing the time allocation parameter, the passive beamforming at each RIS, the transmit beamforming at BS, and the energy partition parameters for all users subject to the size of sensed data, energy supply and given time cycle. The formulated nonconvex problem is tightly coupled by the time allocation parameter and involves the mathematical expectations, which cannot be solved straightly. We use Monte Carlo and fractional programming methods to transform the nonconvex objective function and then propose an alternating optimization-based algorithm to find an approximate solution with guaranteed convergence. Numerical results show that the RIS-aided SR system outperforms other benchmarks in sensing. Furthermore, with the aid of RIS, the channel and system performance can be significantly improved.