The neural network structures of generative models and their corresponding inference models paired in variational autoencoders (VAEs) play a critical role in the models' generative performance. However, powerful VAE network structures are hand-crafted and fixed prior to training, resulting in a one-size-fits-all approach that requires heavy computation to tune for given data. Moreover, existing VAE regularization methods largely overlook the importance of network structures and fail to prevent overfitting in deep VAE models with cascades of hidden layers. To address these issues, we propose a Bayesian inference framework that automatically adapts VAE network structures to data and prevent overfitting as they grow deeper. We model the number of hidden layers with a beta process to infer the most plausible encoding/decoding network depths warranted by data and perform layer-wise dropout regularization with a conjugate Bernoulli process. We develop a scalable estimator that performs joint inference on both VAE network structures and latent variables. Our experiments show that the inference framework effectively prevents overfitting in both shallow and deep VAE models, yielding state-of-the-art performance. We demonstrate that our framework is compatible with different types of VAE backbone networks and can be applied to various VAE variants, further improving their performance.