Thermoresponsive hydrogel-based wound dressings with an incorporated antimicrobial agent can be fabricated employing 3D printing technology. A novel printable ink containing poly(N-isopropylacrylamide) (PNIPAAm) precursors, sodium alginate (ALG), methylcellulose (MC) that is laden with a mixture of octenidine dihydrochloride and 2-phenoxyethanol (Octenisept^®, OCT) possess accurate printability and shape fidelity. This study also provides the protocol of ink’s use for the 3D printing of hydrogel scaffolds. The hydrogel’s physicochemical properties and drug release profiles from the hydrogel specimens to the external solution have been determined at two temperatures (20 and 37 °C). The release test showed a sustained OCT delivery into ultrapure water and the PBS solution. The temperature-responsive hydrogel exhibited antimicrobial activity against Staphylococcus aureus, Candida albicans, and Pseudomonas aeruginosa and demonstrated non-cytotoxicity towards fibroblasts. The thermoresponsive behavior along with biocompatibility, antimicrobial activity, and controlled drug release make this hydrogel a promising class of materials for wound dressing applications.