Biocompatible polymers with the ability to load and release a cargo at the site of action in a smart response to stimuli have attracted great attention in the field of drug delivery and cancer therapy. In this work, we synthesize a dual-responsive dendritic polyglycerol sulfate (DR-dPGS) drug delivery system by copolymerization of glycidol, ε-caprolactone and an epoxide monomer bearing a disulfide bond (SSG), followed by sulfation of terminal hydroxyl groups of the copolymer. The effect of different catalysts, including Lewis acids and organic bases, on the molecular weight, monomer content and polymer structure was investigated. The degradation of the polymer backbone was proven in presence of reducing agents and candida antarctica Lipase B (CALB) enzyme, which results in the cleavage of the disulfides and ester bonds, respectively. The hydrophobic anticancer drug Doxorubicin (DOX) was loaded in the polymer and the kinetic assessment showed an enhanced drug release with glutathione (GSH) or CALB as compared to controls and a synergistic effect of a combination of both stimuli. Cell uptake was studied by using confocal laser scanning microscopy with HeLa cells and showed the uptake of the Dox-loaded carriers and the release of the drug into the nucleus. Cytotoxicity tests with three different cancer cell lines showed good tolerability of the polymers of as high concentrations as 1 mg mL−1, while cancer cell growth was efficiently inhibited by DR-dPGS@Dox.