Artificial intelligence (AI) based downlink channel state information (CSI) prediction for frequency division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems has attracted growing attention recently. However, existing works focus on the downlink CSI prediction for the users under a given environment and is hard to adapt to users in new environment especially when labeled data is limited. To address this issue, we formulate the downlink channel prediction as a deep transfer learning (DTL) problem, where each learning task aims to predict the downlink CSI from the uplink CSI for one single environment. Specifically, we develop the direct-transfer algorithm based on the fully-connected neural network architecture, where the network is trained on the data from all previous environments in the manner of classical deep learning and is then fine-tuned for new environments. To further improve the transfer efficiency, we propose the meta-learning algorithm that trains the network by alternating inner-task and across-task updates and then adapts to a new environment with a small number of labeled data. Simulation results show that the direct-transfer algorithm achieves better performance than the deep learning algorithm, which implies that the transfer learning benefits the downlink channel prediction in new environments. Moreover, the meta-learning algorithm significantly outperforms the direct-transfer algorithm in terms of both prediction accuracy and stability, which validates its effectiveness and superiority.