This work describes the optical design of the EXperiment for Cryogenic Large-Aperture Intensity Mapping (EXCLAIM). EXCLAIM is a balloon-borne telescope that will measure integrated line emission from carbon monoxide (CO) at redshifts z < 1 and ionized carbon ([CII]) at redshifts z = 2.5-3.5 to probe star formation over cosmic time in cross-correlation with galaxy redshift surveys. The EXCLAIM instrument will observe at frequencies of 420--540 GHz using six microfabricated silicon integrated spectrometers with spectral resolving power R = 512 coupled to kinetic inductance detectors (KIDs). A completely cryogenic telescope cooled to a temperature below 5 K provides low-background observations between narrow atmospheric lines in the stratosphere. Off-axis reflective optics use a $90$-cm primary mirror to provide 4.2 full-width at half-maximum (FWHM) resolution at the center of the EXCLAIM band over a field of view of 22.5. Illumination of the 1.7 K cold stop combined with blackened baffling at multiple places in the optical system ensures low (< -40 dB) edge illumination of the primary to minimize spill onto warmer elements at the top of the dewar.
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