Farmers are beginning to promote their agricultural products through e-commerce platform marketplaces beyond conventional retail modes. Platform financing based on this new retail model has gained popularity in e-commerce agricultural supply chains. This study investigates strategic decisions in a dual-channel supply chain, where a capital-constrained farmer can receive funds from an e-commerce platform or a bank under a socially responsible (for-profit) scenario in which the platform does (does not) consider social responsibility. We examine the farmer’s preference between different financing modes (bank versus platform financing) and study the influences of the social responsibility concern on optimal decisions for the farmer and platform and supply chain performance. This study provides some relevant results. First, the farmer faces a trade-off when choosing a financing format: the wholesale price is higher under bank financing, whereas the total planting quantity is higher under platform financing. Second, the comparisons for different financing formats show that when the platform financing rate is relatively low, the farmer and the supply chain prefer platform financing, which is influenced by other significant parameters, such as bank financing rate, commission rate, and social responsibility concern. Third, the social responsibility concern positively influences farmer and supply chain preferences for platform financing. Additionally, the platform’s and supply chain’s profitability and consumer surplus exhibit non-monotonic relationships with the concern about social responsibility, although the farmer’s profit grows with the increased social responsibility concern. Finally, we examine the situation in which the farmer has a non-zero initial capital.