This paper studies the cross-border e-commerce of agricultural products and its logistics supply chain collaborative management approach, the overall transaction mode and basic content, and proposes a cross-border e-commerce supply chain conceptual model. Aiming at the problems of agricultural product supply chains, a method for predicting agricultural product export prices is proposed. The Prophet algorithm under deep learning is utilized to construct the Prophet agricultural product price prediction model for trend, cycle, and holiday terms. Over the introduction of RNN algorithms and LSEM algorithms to optimize the prediction performance of the model, as well as the gradient explosion. On this basis, GRU neural networks are proposed as an improved model of RNN-LSTM. Prediction comparison experiments are designed to empirically analyze agricultural export price prediction and supply chain logistics risk control, and the results of the empirical analysis show that the vegetable export price predicted by using Prophet algorithm during the period of date 2013/4-2013/9 is 2.975, which differs from the actual price by 0.009 yuan, and the margin of error is in the interval of [-0.091,0.014], which is the smallest variation among the three algorithms, which shows that Prophet model has the best performance. After optimizing the FAPSC risk control coefficient, the risk value of supply chain logistics and transportation was successfully reduced from 0.364 to 0.296, and FAPSC effectively minimized the risk.