The yeast strain and storage time is an important factor affecting the development of phytochemicals and sensory attributes in ciders. Therefore, the aim of this study was to determine the influence of yeast strains (Saccharomyces bayanus and Saccharomyces cerevisiae), β-cyclodextrin (BCD), and storage time on physicochemical parameters, contents of phenolic compounds (ultra-performance liquid chromatography with photodiode array detector coupled to quadrupole time-of-flight tandem mass spectrometry (UPLC–PDA–QToF-MS/MS)), antioxidative activity (free radical-scavenging ability (ABTS) and ferric reducing antioxidative power (FRAP) assay), and sensory attributes of new cider from the “Bella Marii” cultivar of red apple. The pH value, acidity, concentrations of alcohol, organic acids, and polyphenols; and the color and antioxidative properties were evaluated in red apple ciders immediately after fermentation and after three months of storage at 4 °C. S. cerevisiae SIHAFERM Finesse Red with BCD and SIHAFERM Finesse Red yeast strain especially contributed to obtaining ciders with a high content of the tested compounds. The use of BCD during fermentation significantly influenced the protection of bioactive compounds, by as much as 18%. Storage time had an impact on concentrations of the tested components (mainly on the total flavan-3-ols and phenolic acids). Based on the achieved values of parameters analyzed in red apple ciders and results of the consumer acceptance test, it may be concluded that red apple offers vast potential for the production of ciders with a high content of polyphenolic compounds.