In recent years, due to an expansion of antibiotic-resistant microorganisms, there has been growing interest in biodegradable and antibacterial polymers that can be used in selected biomedical applications. The present work describes the synthesis of antimicrobial polylactide-copper alginate (PLA–ALG–Cu2+) composite fibers and their characterization. The composites were prepared by immersing PLA fibers in aqueous solution of sodium alginate, followed by ionic cross-linking of alginate chains within the polylactide fibers with Cu(II) ions to yield PLA–ALG–Cu2+ composite fibers. The composites, so prepared, were characterized by scanning electron microscopy (SEM), UV/VIS transmittance and attenuated total reflection Fourier-transform infrared spectroscopy ATR-FTIR, and by determination of their specific surface area (SSA), total/average pore volumes (through application of the 5-point Brunauer–Emmett–Teller method (BET)), and ability to block UV radiation (determination of the ultraviolet protection factor (UPF) of samples). The composites were also subjected to in vitro antimicrobial activity evaluation tests against colonies of Gram-negative (E. coli) and Gram-positive (S. aureus) bacteria and antifungal susceptibility tests against Aspergillus niger and Chaetomium globosum fungal mold species. All the results obtained in this work showed that the obtained composites were promising materials to be used as an antimicrobial wound dressing.