Modern society strives for the development of sustainable processes that are aimed at meeting human needs while preserving the environment. Membrane technologies satisfy all the principles of sustainability due to their advantages, such as cost-effectiveness, environmental friendliness, absence of additional reagents and ease of use compared to traditional separation methods. In the present work, novel green membranes based on sodium alginate (SA) modified by a FeBTC metal–organic framework were developed for isopropanol dehydration using a membrane process, pervaporation. Two kinds of SA-FeBTC membranes were developed: (1) untreated membranes and (2) cross-linked membranes with citric acid or phosphoric acid. The structural and physicochemical properties of the developed SA-FeBTC membranes were studied by spectroscopic techniques (FTIR and NMR), microscopic methods (SEM and AFM), thermogravimetric analysis and swelling experiments. The transport properties of developed SA-FeBTC membranes were studied in the pervaporation of water–isopropanol mixtures. Based on membrane transport properties, 15 wt % FeBTC was demonstrated to be the optimal content of the modifier in the SA matrix for the membrane performance. A membrane based on SA modified by 15 wt % FeBTC and cross-linked with citric acid possessed optimal transport properties for the pervaporation of the water–isopropanol mixture (12–100 wt % water): 174–1584 g/(m2 h) permeation flux and 99.99 wt % water content in the permeate.