Adapting to the changing climate requires accurate local climate information, a computationally challenging problem. Recent studies have used Generative Adversarial Networks (GANs), a type of deep learning, to learn complex distributions and downscale climate variables efficiently. Capturing variability while downscaling is crucial for estimating uncertainty and characterising extreme events - critical information for climate adaptation. Since downscaling is an undetermined problem, many fine-scale states are physically consistent with the coarse-resolution state. To quantify this ill-posed problem, downscaling techniques should be stochastic, able to sample realisations from a high-resolution distribution conditioned on low-resolution input. Previous stochastic downscaling attempts have found substantial underdispersion, with models failing to represent the full distribution. We propose approaches to improve the stochastic calibration of GANs in three ways: a) injecting noise inside the network, b) adjusting the training process to explicitly account for the stochasticity, and c) using a probabilistic loss metric. We tested our models first on a synthetic dataset with known distributional properties, and then on a realistic downscaling scenario, predicting high-resolution wind components from low-resolution climate covariates. Injecting noise, on its own, substantially improved the quality of conditional and full distributions in tests with synthetic data, but performed less well for wind field downscaling, where models remained underdispersed. For wind downscaling, we found that adjusting the training method and including the probabilistic loss improved calibration. The best model, with all three changes, showed much improved skill at capturing the full variability of the high-resolution distribution and thus at characterising extremes.