Novel nanofibrous membranes with outstanding adsorption property and high durability were fabricated by chemically immobilizing carboxymethyl-beta-cyclodextrin polymers on the surfaces of biodegradable poly((butylene succinate-co- terephthalate)-co-serinol terephthalate) nanofibrous membranes via the amidation reaction. As an ecofriendly adsorbent, the resulting nanofibrous membranes presented rapid adsorption toward methylene blue (MB) dye with a removal efficiency of about 90% within 30 s and had the maximum adsorption capacity of 1030.9 mg/g to it. The thermodynamic parameters indicated that the adsorption process of the adsorbent to MB dye was endothermic and spontaneous. Selective adsorption was another advantage to the adsorbent, which showed preferable adsorption to cationic dye MB rather than anionic dye methyl orange with a separation efficiency of 99.9% to them. Inspiringly, the removal efficiency and mechanical properties of the adsorbent hardly changed after 20 regeneration cycles, indicating its good regeneration and high durability in treating the dye-containing wastewater.