Internet of Things (IoT)-enabled infrastructures have naturally a multi-layer system architecture with an overlaid or underlaid device network and its coexisting infrastructure network. The connectivity between different components in these two heterogeneous networks plays an important role in ensuring a high-level situational awareness. However, IoTenabled infrastructures face cyber threats due to the wireless nature of communications. Therefore, maintaining the network connectivity in the presence of adversaries is a critical task for the infrastructure network operators. In this paper, we establish a three-player three-stage game-theoretic framework including two network operators and one attacker to capture the secure design of multi-layer infrastructure networks by allocating limited resources. We use subgame perfect Nash equilibrium (SPE) to characterize the strategies of players with sequential moves. In addition, we assess the efficiency of the equilibrium network by comparing with its team optimal solution counterparts in which two network operators can coordinate. We further design a scalable algorithm to guide the construction of the equilibrium IoT-enabled infrastructure networks. Finally, we use case studies on the emerging paradigm of Internet of Battlefield Things (IoBT) to corroborate the obtained results.