The growing electricity demand of AI data centers introduces significant voltage variability in power networks, affecting not only their own operation but also the experience of all users sharing the network. To smooth data center impacts on power networks, we develop an online feedback optimization approach that controls distributed battery energy storage systems to mitigate voltage issues induced by data center operations. The controller adjusts the active and reactive power setpoints of distributed battery systems in response to voltage measurements, with a two-fold objective: managing voltage to minimize the magnitude of constraint violations and smoothing voltage profiles. Control performance is evaluated in a high-fidelity simulation environment that integrates a three-phase distribution feeder and a detailed battery system model, and benchmarked against a local control approach with similar objectives but without optimality guarantees and constraint enforcement. We show that the proposed controller delivers consistent voltage regulation in the long term, while the local control approach pursues the objectives more aggressively but quickly hits the storage limits.