Wines represent a rich source of bioactive compounds, especially polyphenolic compounds, which mostly contribute to the antioxidant activity. Sulfur dioxide (SO2) has mostly been used as a preservative in winemaking to prevent oxidation during storage. The aim of this paper is to evaluate the changes in SO2 levels and the influence of sulfur dioxide addition at seven different concentrations on the antioxidant activity (detected by DPPH-2,2-diphenyl-1-picrylhydrazyl and the ABTS Trolox equivalent antioxidant capacity methods), total polyphenols, flavonoid content and individual polyphenolic compounds (determined by the HPLC high-performance liquid chromatography method) of white wines during 5 months of storage. The assayed sulfur dioxide concentrations show a decreasing tendency with time, with a final decrease of more than 50% in comparison with the start of the experiment. Between the first and second measurements, the average decrease in sulfur dioxide was 16%. In the following interval, it was found that the maximum decline in SO2 was 26%. The changes in SO2 levels cannot be considered statistically significant. At the same time, we observed a decreasing tendency in the TPC content during storage. The antioxidant activity determined by the DPPH method at the beginning of the experiment ranged from 116.75 up to 270.62 mg·L−1, while the antioxidant activity increased with sulfur dioxide concentration. The AA detected by the ABTS method displayed a decreasing tendency during storage. In the case of the TFC content, we observed a significant influence of sulfur dioxide on the concentration. No addition or addition of high SO2 concentrations negatively influenced the flavonoid content in the samples. During storage, we observed a highly variable content of phenolic compounds in relation to SO2 addition. The most abundant compounds were chlorogenic acid, caffeic acid and epigallocatechin.