Climate downscaling is a crucial technique within climate research, serving to project low-resolution (LR) climate data to higher resolutions (HR). Previous research has demonstrated the effectiveness of deep learning for downscaling tasks. However, most deep learning models for climate downscaling may not perform optimally for high scaling factors (i.e., 4x, 8x) due to their limited ability to capture the intricate details required for generating HR climate data. Furthermore, climate data behaves differently from image data, necessitating a nuanced approach when employing deep generative models. In response to these challenges, this paper presents a deep generative model for downscaling climate data, specifically precipitation on a regional scale. We employ a denoising diffusion probabilistic model (DDPM) conditioned on multiple LR climate variables. The proposed model is evaluated using precipitation data from the Community Earth System Model (CESM) v1.2.2 simulation. Our results demonstrate significant improvements over existing baselines, underscoring the effectiveness of the conditional diffusion model in downscaling climate data.