Artificial Intelligence (AI) is rapidly becoming a foundational layer of social, economic, and cognitive infrastructure. At the same time, the training and large-scale deployment of AI systems rely on finite and unevenly distributed energy, networking, and computational resources. This tension exposes a largely unexamined problem in current AI governance: while expanding access to AI is essential for social inclusion and equal opportunity, unconstrained growth in AI use risks unsustainable resource consumption, whereas restricting access threatens to entrench inequality and undermine basic rights. This paper argues that access to AI outputs largely derived from publicly produced knowledge should not be treated solely as a commercial service, but as a fundamental civil interest requiring explicit protection. We show that existing regulatory frameworks largely ignore the coupling between equitable access and resource constraints, leaving critical questions of fairness, sustainability, and long-term societal impact unresolved. To address this gap, we propose recognizing access to AI as an \emph{Intergenerational Civil Right}, establishing a legal and ethical framework that simultaneously safeguards present-day inclusion and the rights of future generations. Beyond normative analysis, we explore how this principle can be technically realized. Drawing on emerging paradigms in IoT--Edge--Cloud computing, decentralized inference, and energy-aware networking, we outline technological trajectories and a strawman architecture for AI Delivery Networks that support equitable access under strict resource constraints. By framing AI as a shared social infrastructure rather than a discretionary market commodity, this work connects governance principles with concrete system design choices, offering a pathway toward AI deployment that is both socially just and environmentally sustainable.