The ever-increasing tendency toward economic globalization highlights the importance of sustainable container transport networks to a country’s international trade, especially for an economy that is highly dependent on exports. This paper aims to develop a transport network connectivity index (TNCI) to measure the container transport connectivity from a multi-modal perspective. The proposed index is based on both graph theory and economics, considering transport infrastructure and capacity, cargo flow, and capacity utilization. Using the case of South Korea as an example, we apply the TNCI to assess the connectivity of the Busan, Gwangyang, and Incheon ports, representing approximately 96% of the container throughput in South Korea. The calculated TNCI not only provides insight into the assessment of sustainable port competitiveness, it also helps policymakers identify bottlenecks in multi-modal transport networks. To eliminate these bottlenecks, this paper offers some appropriate measures and specific strategies for port development, which in turn improves the connectivity of container transport networks for sustainable development.