The availability of reliable, high-resolution climate and weather data is important to inform long-term decisions on climate adaptation and mitigation and to guide rapid responses to extreme events. Forecasting models are limited by computational costs and therefore often can only make coarse resolution predictions. Statistical downscaling can provide an efficient method of upsampling low-resolution data. In this field, deep learning has been applied successfully, often using image super-resolution methods from computer vision. Despite achieving visually compelling results in some cases, such models often violate conservation laws when predicting physical variables. In order to conserve important physical quantities, we develop methods that guarantee physical constraints are satisfied by a deep downscaling model while also increasing their performance according to traditional metrics. We introduce two ways of constraining the network: A renormalization layer added to the end of the neural network and a successive approach that scales with increasing upsampling factors. We show the applicability of our methods across different popular architectures and upsampling factors using ERA5 reanalysis data.