Recent progress in the accurate calculation of noncovalent interactions has enabled density-functional theory (DFT) to model systems relevant in biological and supramolecular chemistry. The application of DFT methods using atom-centered Gaussian basis sets to large systems is limited by the number of basis functions required to accurately model thermochemistry and, in particular, weak intermolecular interactions. Basis set incompleteness error (BSIE) arising from the use of incomplete basis sets leads to erroneous intermolecular energies, bond dissociation energies, and structures. In this article, we develop a correction for BSIE in DFT calculations using basis set incompleteness potentials (BSIP). BSIPs are atom-based one-electron potentials (ACPs) with the same functional form as effective core potentials (ECP) that are designed to correct the effects of BSIE in properties that are linear mappings of the energy. We present a systematic way of developing general, error-correcting ACPs and apply this technique to generate BSIPs for eight common elements in organic and biological systems (H, C, N, O, F, P, S, and Cl). Two BSIPs were optimized for use with the scaled MINI (MINIs) and MINIs(d) basis sets and were designed to correct for the impacts of BSIE on noncovalent binding energies and intra/intermolecular geometries. BSIPs developed for use with 6-31G*, pc-1, and 6-31+G** basis sets also correct for the effects of BSIE on bond dissociation energies, which enables the study of chemical reactions in very large systems. BSIPs can be used with any density functional in any electronic structure program that implements ECPs. Our BSIPs add very little to the computational cost provided an efficient ECP implementation is used. Our results support the use of BLYP-D3/MINIs-BSIP as a computationally inexpensive and more accurate alternative to other approaches (e.g., B3LYP/6-31G* and BP86/6-31G*) in protein and supramolecular structural studies.