With the development of the Internet, social network platforms (SNPs) have become the most common channel for image sharing. As a result, transmitting stego images in the public channels gives steganographers the best opportunity to transmit secret messages with behavioral security preserved. However, the SNPs typically compress uploaded images and damage the weak signal of steganography. In this study, a robust JPEG steganographic scheme based on robustness measurement and cover block selection (CBSRS) is proposed. We first design a deep learning-based model to fit the blockwise change rate of coefficients after JPEG recompression. Then, a cover block selection strategy is proposed to improve the robustness by optimizing the joint distortion function of transmission costs and classic costs. Moreover, by embedding indicator of cover block selection in chrominance channels of JPEG images, a shareable cover construction scheme is designed to solve the problem of auxiliary information transmission. The experimental results show that our proposed framework improves robustness while maintaining statistical security. Comparing with state-of-the-art methods, the framework achieves better performance under given recompression channels.