Cross-modal hashing has become a vital technique in cross-modal retrieval due to its fast query speed and low storage cost in recent years. Generally, most of the priors supervised cross-modal hashing methods are flat methods which are designed for non-hierarchical labeled data. They treat different categories independently and ignore the inter-category correlations. In practical applications, many instances are labeled with hierarchical categories. The hierarchical label structure provides rich information among different categories. To rationally take use of category correlations, hierarchical cross-modal hashing is proposed. However, existing methods intend to preserve instance-pairwise or class-pairwise similarities, which cannot fully explore the semantic correlations among different categories and make the learned hash codes less discriminative. In this paper, we propose a deep cross-modal hashing method named hierarchical semantic structure preserving hashing (HSSPH), which directly exploits the label hierarchy information to learn discriminative hash codes. Specifically, HSSPH learns a set of class-wise hash codes for each layer. By augmenting class-wise codes with labels, it generates layer-wise prototype codes which reflect the semantic structure of each layer. In order to enhance the discriminative ability of hash codes, HSSPH supervises the hash codes learning with both labels and semantic structures to preserve the hierarchical semantics. Besides, efficient optimization algorithms are developed to directly learn the discrete hash codes for each instance and each class. Extensive experiments on two benchmark datasets show the superiority of HSSPH over several state-of-the-art methods.