In this study, we used electrospinning to produce core-shell nanofibers of poly(lactic acid) as core and polyacrylonitrile/cellulose nanocrystals (CNC) or polyacrylonitrile/chitin nanocrystals (ChNC) as shell. Electrospun materials prepared at different nanocrystal concentrations were tested and assayed as microfiltration membranes. The coaxial membranes presented a maximum pore size in the 1.2–2.6 μm range and rejections > 85% for bacterial cells (0.5 × 2.0 μm) and > 99% for fungal spores (> 2 μm). The morphological and mechanical properties and the water permeability of the nanocomposite membranes were studied. The morphological characterization showed random fibers of beadless and well-defined core/shell structured fibers with diameter generally below the micron size with presence of secondary ultrafine nanofibers. Tensile strength and Young's modulus of elasticity improved with respect to coaxial membranes without nanocrystals with best mechanical properties achieved at 5 wt% CNC and 15 wt% ChNC loadings. The enhancement was attributed to the reinforcing effect of the percolating network of cellulose nanocrystals. Water permeability increased for all membranes loaded with nanocrystals with respect to the coaxial fibers without nanocrystals, the highest corresponding to ChNC composites with up to a 240% increase over non-loaded membranes. Composite membranes prepared with CNC in their shell were hydrophilic, in contrast with the hydrophobic PLA core, while coaxial fibers with ChNC were superhydrophilic. CNC membranes were negatively charged but ChNC originated neutral or positively charged membranes due to the contribution of deacetylated chitin structural units. Upon exposure to E. coli cultures, composite membranes containing ChNC showed a high antimicrobial action and were essentially free of bacterial colonization under strong biofilm formation conditions.