The programmable, sequence-dependent hybridization of DNA has spurred the development of DNA hydrogels, polymer networks that swell in water and are comprised either entirely or partially of DNA. Specific applications require hydrogels of particular structure for optimal functionality. Here, we use self-assembling, multi-valent DNA nanostars to examine how hydrogel structure is influenced by the non-equilibrium dynamics of its interacting components. We show that hydrogel aging kinetics - from an arrested, solid-like percolated network to an equilibrium, phase-separated liquid analogous to coacervates - are modulated by DNA hybridization strength and ion-specific nanostar internal flexibility. Together, our results demonstrate strategies to control hydrogel kinetic phenomena, and thus the hydrogel structure, through the rational design of gel-forming elements and solvent conditions.