With the rapid advancement of the Internet of Things (IoT), the typical application of wireless body area networks (WBANs) based smart healthcare has drawn wide attention from all sectors of society. To alleviate the pressing challenges, such as resource limitations, low-latency service provision, mass data processing, rigid security demands, and the lack of a central entity, the advanced solutions of fog computing, software-defined networking (SDN) and blockchain are leveraged in this work. On the basis of these solutions, a task offloading strategy with a centralized low-latency, secure and reliable decision-making algorithm having powerful emergency handling capacity (LSRDM-EH) is designed to facilitate the resource-constrained edge devices for task offloading. Additionally, to well ensure the security of the entire network, a comprehensive blockchain-based two-layer and multidimensional security strategy is proposed. Furthermore, to tackle the inherent time-inefficiency problem of blockchain, we propose a blockchain sharding scheme to reduce system time latency. Extensive simulation has been conducted to validate the performance of the proposed measures, and numerical results verify the superiority of our methods with lower time-latency, higher reliability and security.