We present Infrared Space Observatory (ISO) measurements of the far-infrared (FIR) fine-structure lines [S IV] (10.5 mu m), [S III] (33.5 mu m), [O III] (51.8 and 88.4 mu m), [N III] (57.3 mu m), [N II] (121.9 mu m), [O I] (63.2 and 145.5 mu m), and [C II] (157.7 mu m) at five positions on the thermal "Arched'' Filaments in the Galactic center region in order to investigate the influence of the massive, young Arches cluster on the interstellar medium. Serendipitous measurements of OH (119.3 mu m) are also presented. Radio recombination line and ISO spectroscopic measurements provide strong evidence that the Arches cluster is responsible for the excitation of the Arched Filaments. The observations reported here were selected to provide measurements close to the Arches cluster and at uniformly increasing projected distances over the range 0-9 pc. Within the measurement uncertainties, the diagnostic line ratios most sensitive to excitation and least sensitive to density fluctuations, [O III]/[S III] and [S IV]/[S III], appear to decrease with projected distance from the Arches cluster. In agreement with other studies, this provides additional evidence that the cluster is the primary ionization source for the Arched Filaments. The total FIR luminosity also generally falls with projected distance from the Arches cluster. The best predictions from photoionization models for the rapid decrease in the measured line flux ratios with projected distance from the Arches cluster, however, are only accurate to a factor of 1.5, including allowance for additional offset along the line of sight. The measurements of lines formed predominantly in photodissociation regions are not well matched by current models. High-resolution spectra indicate gas moving at -70 km s(-1) only at the location of the Arches cluster, significantly different from the predominant -30 km s(-1) gas seen in the underlying molecular cloud.