Railway catenary galloping, induced by aerodynamic instability, poses a significant threat by disrupting the electric current connection through sliding contact with the contact wire. This disruption leads to prolonged rail service interruptions and damage to the catenary’s suspension components. This paper delves into the exploration of optimizing the catenary system’s structure to alleviate galloping responses, addressing crucial parameters such as span length, stagger dropper distribution, and tension levels. Employing a finite element model, the study conducts simulations to analyze the dynamic response of catenary galloping, manipulating structural parameters within specified ranges. To ensure accurate and comprehensive exploration, the Sobol sequence is utilized to generate low-discrepancy, quasi-random, and super-uniform distribution sequences for the high-dimensional parameter inputs. Subsequent to the simulation phase, a genetic algorithm based on neural networks is employed to identify optimal parameter settings for suppressing catenary galloping, taking into account various constraints. The results gleaned from this investigation affirm that adjusting structural parameters can effectively diminish the galloping amplitude of the railway catenary. The most impactful strategy involves augmenting tension and reducing span length. Moreover, even when tension and span length are fixed, adjusting other parameters demonstrates efficacy in reducing galloping amplitudes. The adjustment of messenger-wire tension, dropper distribution, and stagger can achieve a 22.69% reduction in the maximum vertical galloping amplitude. Notably, maintaining a moderate stagger value and a short steady arm–dropper distance is recommended to achieve the minimum galloping amplitude. This research contributes valuable insights into the optimization of railway catenary systems, offering practical solutions to mitigate galloping-related challenges and enhance overall system reliability.