In this paper we study the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with correlated noise by field-theoretic dynamic renormalization-group techniques. We focus on spatially correlated noise where the correlations are characterized by a sinc profile in Fourier space with a certain correlation length ξ. The influence of this correlation length on the dynamics of the KPZ equation is analyzed. It is found that its large-scale behavior is controlled by the standard KPZ fixed point, i.e., in this limit the KPZ system forced by sinc noise with arbitrarily large but finite correlation length ξ behaves as if it were excited by pure white noise. A similar result has been found by Mathey et al. [S. Mathey et al., Phys. Rev. E 95, 032117 (2017)2470-004510.1103/PhysRevE.95.032117] for a spatial noise correlation of Gaussian type (∼e^{-x^{2}/2ξ^{2}}), using a different method. These two findings together suggest that the KPZ dynamics is universal with respect to the exact noise structure, provided the noise correlation length ξ is finite.