The new sandwich composite structure formed by basalt fiber resin-based sheets and Nomex honeycomb has the advantages of being lightweight and environmentally friendly, as well as having excellent electromagnetic performance. It has very important application prospects in traditional and emerging fields. In this paper, the mechanical properties of this new sandwich composite structure are studied. The results show that, under the condition of flatwise compression, increasing the height of the honeycomb is conducive to improving the compressive capacity of the structure. However, the height should be controlled within a certain range in case of instability and yield of the honeycomb; under the bending conditions, the bending failure mode of the composite structure has gone through five stages. Owing to the honeycomb manufacturing process, the orientation of the honeycomb also has a great influence on the bending strength of the structure. After further analysis, it is found that basalt fiber sheets contribute the most to the bending stiffness of the structure, and the main role of honeycomb is to provide out-of-plane support. In both cases, the failure of specimens is ductile, and the combined structure still has a small amount of bearing capacity and maintains structural integrity. Research on this new type of composite structural material is very beneficial for promoting the application and development of green and lightweight special functional materials.