Controlled drug release systems are the subject of many investigations to achieve the therapeutic effect of drugs. They have numerous advantages, such as localized effects, lower side effects, and less onset of action. Among drug-delivery systems, electrospinning is a versatile and cost-effective method for biomedical applications. Furthermore, electrospun nanofibers are promising as drug carrier candidates due to their properties that mimic the extracellular matrix. In this work, electrospun fibers were made of Poly-L-lactic acid (PLA), one of the most widely tested materials, which has excellent biocompatible and biodegradable properties. A curcuminoid, bisdemethoxycurcumin (BDMC) was added in order to complete the drug delivery system. The PLA/BDMC membranes were characterized, and biological characteristics were examined in vitro. The results show that the average fiber diameter was reduced with the drug, which was mainly released during the first 24 h by a diffusion mechanism. It was seen that the use of our membranes loaded with BDMC enhanced the rate of proliferation in Schwann cells, the main peripheral neuroglial cells, and modulated inflammation by reducing NLRP3 inflammasome activation. Considering the results, the prepared PLA/BDMC membranes hold great potential for being used in tissue engineering applications.