In this study, poly(vinyl alcohol) (PVA) membranes containing Ag nanoparticles (AgNPs) were prepared by electrospinning and grafted copolymerization with 3,3′,4,4′-benzophenone tetracarboxylic acid (BPTA) to provide better mechanical properties, lower water vapor transmittance, and higher antibacterial activity (against Staphylococcus aureus and Escherichia coli) than the PVA/AgNPs membrane. The PVA/AgNPs/BPTA membrane showed higher antibacterial activity than the other membranes, and it produced inhibition zones with diameters of 18.12 ± 0.08 and 16.41 ± 0.05 mm against S. aureus and E. coli, respectively. The PVA/AgNPs/BPTA membrane was found to be capable of promoting reactive oxygen species (ROS) formation under both light and dark conditions. Cycling experiments performed following ROS quenching showed that the best-performing composite membrane retained >70% of its original OH⋅ radical and H2O2 charging capacity after seven cycles. In the filtration test, the electrospun nanofibrous membranes showed high filtration efficiencies of 99.98% for sodium chloride (NaCl). In addition, these membranes maintained a relatively low pressure drop of 168 Pa with a basis weight of 2.1 g m−2. Thus, the PVA/AgNPs/BPTA membrane was concluded to be a promising medical protective material offering the benefits of structural stability and reusability.