Binding to RNA has been observed for an ever-increasing number of proteins, which often have other functions. The contributions of RNA binding to protein function are best discerned by studying separation-of-function mutants that hamper interaction with RNA without affecting other aspects of protein function. To design these mutants, we need precise knowledge of the residues that contribute to the affinity of the protein for its RNA ligands. Here, we present RBR-scan: a technology to simultaneously measure RNA-binding affinity of a large number of protein variants. We fused individual variants with unique peptide barcodes optimized for detection by mass spectrometry (MS), purified protein pools from single bacterial culture, and assayed proteins in parallel for RNA binding. Mutations in the MS2 coat protein known to impair RNA-binding were correctly identified, as well as a previously unreported mutant, which we validated with orthogonal biochemical methods. We used RBR-scan to discover novel RNA-binding mutants in the cancer-associated splicing regulator SRSF2. Together, our results demonstrate that RBR-scan is a powerful and scalable platform for linking RNA-binding affinity to protein sequence, offering a novel strategy to decode the functional consequences of protein-RNA interactions.