The emerging Internet of Things (IoT) applications that leverage ubiquitous connectivity and big data are facilitating the realization of smart everything initiatives. IoT-enabled infras-tructures can be naturally divided into two layers including the existing infrastructure layer and the underlaid device network. The connectivity between different components in the infrastructure networks plays an important role in delivering real-time information and ensuring a high-level situational awareness. However, IoT-enabled 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 to design a secure network. We further design a scalable algorithm to construct the equilibrium secure IoT-enabled infrastructure networks. Finally, we use case studies on Internet of Battlefield Things (IoBT) to corroborate the obtained results.