Upon stress, profound post-transcriptional adjustments of gene expression occur in spatially restricted, subcellular, membraneless compartments, or ribonucleoprotein (RNP) granules, which are formed by liquid phase separation of RNA-binding proteins with low complexity sequence domains (LCDs). Here, we show that Rbfox1 is an LCD-containing protein that aggregates into liquid droplets and amyloid-like fibers and promiscuously joins different nuclear and cytoplasmic RNP granules. Using Drosophila oogenesis as an in vivo system for stress response, we demonstrate a mechanism by which Rbfox1 promotes cell survival. The stress-dependent miRNA miR-980 acts to buffer Rbfox1 levels, since it targets only those Rbfox1 transcripts that contain extended 3'UTRs. Reduced miR-980 expression during stress leads to increased Rbfox1 levels, widespread formation of various RNP granules, and increased cell viability. We show that human RBFOX proteins also contain multiple LCDs and form membraneless compartments, suggesting that the RNP granule-linked control of cellular adaptive responses may contribute to a wide range of RBFOX-associated pathologies in humans.