A closed-loop supply chain (CLSC) is the process of adding a reverse supply chain to the traditional forward supply chain. It has a positive effect on reducing environmental pollution, and therefore many governments have introduced relevant policies to encourage enterprises to develop CLSCs. Among them, the reward–penalty policy (RPP) for the recovery rate of waste electrical and electronic equipment (WEEE) is considered better. At the corporate level, the development of CLSCs can enhance the social responsibility awareness of the enterprise and effectively enhance the corporate image. Therefore, in order to study the decision making and recycling channel selection of CLSCs that consider corporate social responsibility (CSR) under the government’s RPP, this paper constructs a two-stage CLSC, including a manufacturer, a retailer, a third-party recycler, and the government. By adopting the Stackelberg game method, we determine the optimal results of the three models and draw a series of conclusions: (1) when the manufacturer is responsible for recycling WEEE, the profit is the largest, and it shows an increasing trend as the government’s reward–penalty coefficient increases; (2) the manufacturer recycling model enables the manufacturer to obtain the maximum CSR investment level, followed by the third -party recycler recycling model, and the smallest is the retailer recycling model; and (3) the government tends to build third-party recycling agencies, which is conducive to improving the level of contribution to GDP. The conclusion of this paper has certain management inspirations for the decision making of the enterprise and the formulation of government policies.