Literature reviews have described the applications of silver, copper, and zinc ions and metallic particles of Cu, Ti, and Zn oxides, which have been found to be useful antimicrobial reagents for the biofunctionalization of various materials and their surfaces. For this purpose, compositions of water dispersions containing emulsions of synthetic copolymers based on acrylic and vinyl monomers, polysaccharides (hydroxyethyl cellulose and starch), and various additives with wetting and stabilizing properties were used. Many stable water dispersions of different chemical compositions containing bioactive chemical compounds (copper silicate hydrate, titanium dioxide, and zinc oxide (and other auxiliary substances)) were developed. They were used for the preparation of thin hybrid coatings having good antimicrobial properties against Gram-negative bacteria (Escherichia coli), Gram-positive bacteria (Staphylococcus aureus), and yeast fungus (Candida albicans). Polyester (PES) and polylactide (PLA) nonwovens were modified using the dip-coating method, while PES and cotton fabrics were biofunctionalized by means of dip-coating and coating methods. The antimicrobial (antibacterial and antifungal) properties of the textile materials (nonwovens and fabrics) biofunctionalized with the above-mentioned bioactive agents exhibiting antimicrobial properties (CuSiO3, TiO2, ZnO, or ZnO∙SiO2) were strongly dependent on the agents’ content in the water dispersions. The PES and PLA nonwovens, modified on the surface with water compositions containing copper silicate hydrate, showed good antibacterial properties against the Gram-negative bacteria Escherichia coli, even at a content of 1 wt.% CuSiO3∙xH2O, and against the Gram-positive bacteria Staphylococcus aureus, at the content of at least 5 wt.% CuSiO3∙xH2O. The bacterial growth reduction factor (R) was greater than 99% for most of the samples tested. Good antifungal properties against the fungus Candida albicans were found for the PES and PLA nonwoven fabrics modified with dispersions containing 5–7 wt.% CuSiO3∙xH2O and 4.2–5.0 wt.% TiO2. The addition of TiO2 led to a significant improvement in the antifungal properties of the PES and PLA nonwovens modified in this way. For the samples of PES WIFP-270 and FS F-5 nonwovens, modified with water dispersions containing 5.0 wt.% CuSiO3∙xH2O and 4.2–5.0 wt.% TiO2, the growth reduction factor for the fungus Candida albicans (R) reached values in the range of 80.9–98.0%. These new biofunctionalized polymeric nonwoven textile materials can find practical applications in the manufacture of filters for hospital air-conditioning systems and for the automotive industry, as well as in air purification devices. Moreover, similar antimicrobial modification of fabrics with the dip-coating or coating methods can be applied, for example, in the fabrication of fungi- and mold-resistant garden furniture.