Low-altitude communications can promote the integration of aerial and terrestrial wireless resources, expand network coverage, and enhance transmission quality, thereby empowering the development of sixth-generation (6G) mobile communications. As an enabler for low-altitude transmission, 3D channel fingerprints (3D-CF), also referred to as the 3D radio map or 3D channel knowledge map, are expected to enhance the understanding of communication environments and assist in the acquisition of channel state information (CSI), thereby avoiding repeated estimations and reducing computational complexity. In this paper, we propose a modularized multimodal framework to construct 3D-CF. Specifically, we first establish the 3D-CF model as a collection of CSI-tuples based on Rician fading channels, with each tuple comprising the low-altitude vehicle's (LAV) positions and its corresponding statistical CSI. In consideration of the heterogeneous structures of different prior data, we formulate the 3D-CF construction problem as a multimodal regression task, where the target channel information in the CSI-tuple can be estimated directly by its corresponding LAV positions, together with communication measurements and geographic environment maps. Then, a high-efficiency multimodal framework is proposed accordingly, which includes a correlation-based multimodal fusion (Corr-MMF) module, a multimodal representation (MMR) module, and a CSI regression (CSI-R) module. Numerical results show that our proposed framework can efficiently construct 3D-CF and achieve at least 27.5% higher accuracy than the state-of-the-art algorithms under different communication scenarios, demonstrating its competitive performance and excellent generalization ability. We also analyze the computational complexity and illustrate its superiority in terms of the inference time.