The paper presents the results of an investigation of the nanostructure, elements, and phase composition of thin (100–140 nm) tin dioxide films obtained via magnetron sputtering and containing Ag, Y, Sc, Ag + Y, and Ag + Sc additives in the volume. Electrical and gas-sensitive characteristics of hydrogen sensors based on these films with dispersed Pt/Pd layers deposited on the surface were studied. The additives had a significant effect on the nanostructure of the films, the density of oxygen adsorption sites on the surface of tin dioxide, the band bending at the grain boundaries of tin dioxide, the resistance values in pure air, and the responses to hydrogen in the concentration range of 50–2000 ppm. During the long-term tests of most of the samples studied, there was an increase in the resistance of the sensors in clean air and in the response to hydrogen. It has been established that the joint introduction of Ag + Y into the volume of films prevents the increase in the resistance and response. For these sensors based on thin films of Pt/Pd/SnO2:Sb, Ag, Y the responses to 100 and 1000 ppm of H2 are 25 and 575, correspondingly, the response time at exposure to 100 and 1000 ppm of H2 are 10 and 90 s, the recovery time at exposure to 100 and 1000 ppm of H2 17 and 125 s. Possible mechanisms of the effect of additives on the properties of sensors and the stability of their parameters during long-term operation were considered.