Coaxial electrospinning is a commonly used technique in pharmaceutical sciences that allows producing polymeric nano- or microfibers with core-shell structure. In addition to process parameters such as voltage and flow rate of the polymer solutions, the properties of such fibers are influenced by solution properties including the total polymer concentration and the polymer ratio between fiber core and shell. As these parameters are typically changed empirically during development of electrospinning methods, there is a lack of structured studies that determine the optimal settings for the production of homogeneous, mechanically stable fibers. Therefore, we used a design of experiments approach to systematically address the coaxial electrospinning process of polycaprolactone, a polymer for biomaterials applied to the skin, for instance in the form of patches for wound healing or the treatment of skin cancer. We set out to investigate the influence of different process parameters on the fiber morphology (diameter and size distribution) as well as their mechanical properties as these properties are decisive determinants for handling and pharmaceutical efficacy of such patches. The investigated process parameters were the polymer concentration in the electrospinning solution and the ratio of polymer between fiber core and shell. We additionally investigated the impact of the presence of the model drug hydrocortisone in the fiber core on the electrospinning outcome and the drug release. The experiments revealed that the fiber diameter increases with increasing total polymer concentration, correlating with a narrower fiber size distribution. In addition, our results show that a higher ratio of polymer content in core and shell leads to more mechanically stable fibers (higher tensile strength, higher Young’s modulus). Taken together, the results obtained in this study contribute to a better understanding of the electrospinning process of polycaprolactone.