Curcumin extracted from the rhizome of Curcuma Longa has been used in therapeutic preparations for centuries in different parts of the world. However, its bioactivity is limited by chemical instability, water insolubility, low bioavailability, and extensive metabolism. In this study, the coaxial electrospinning technique was used to produce both poly (ε-caprolactone) (PCL)–curcumin and core–shell nanofibers composed of PCL and curcumin in the core and poly (lactic acid) (PLA) in the shell. Morphology and physical properties, as well as the release of curcumin were studied and compared with neat PCL, showing the formation of randomly oriented, defect-free cylindrical fibers with a narrow distribution of the dimensions. The antibacterial and antibiofilm potential, including the capacity to interfere with the quorum-sensing mechanism, was evaluated on Pseudomonas aeruginosa PAO1, and Streptococcus mutans, two opportunistic pathogenic bacteria frequently associated with infections. The reported results demonstrated the ability of the Curcumin-loading membranes to inhibit both PAO1 and S. mutans biofilm growth and activity, thus representing a promising solution for the prevention of biofilm-associated infections. Moreover, the high biocompatibility and the ability to control the oxidative stress of damaged tissue, make the synthesized membranes useful as scaffolds in tissue engineering regeneration, helping to accelerate the healing process.