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.