Currently developing on a large scale, the opportunities for 3D printing represent more and more perspective solutions in the area of tissue engineering and personalized medicine. Due to their ability to reproduce the natural extracellular matrix and unique properties, hydrogels are popularly used materials to produce bioinks designated for 3D printing. Today, solutions based on sodium alginate and gelatin are frequently used compositions for this purpose. The high viability of the cells incorporated into bioink is the key parameter determining the application opportunities of printed structures. The parameters of the process used for the preparation of hydrogel compositions may have a direct impact on the viability of the cells incorporated within the printed structure. This study aims to develop a protocol for the preparation of hydrogel materials based on alginate and gelatin, providing the highest viability of the model osteoblast-like cell line Saos-2 incorporated directly into the bioink before the 3D bioprinting process. In the scope of this study, the analyzed process parameters of the preparation of the hydrogel bioinks are the method of combination of a polymer solution with biological material, the applied concentration, the cross-linking solution, and also the waiting time of the prepared hydrogel bioink for the 3D printing process. A key aspect of the study is the evaluation of the influence of 3D printing on changes in the survival rate of biological material directly after the manufacturing process and after individual incubation periods of the printouts in conditions reflecting the body’s environment.