A ROSAT HRI X-ray image was obtained of the open globular cluster NGC 288, which is located near the South Galactic Pole. This is the first deep X-ray image of this system. We detect a low-luminosity globular cluster X-ray source (LLGCX) RX J005245.0-263449 with an X-ray luminosity of (5.5 +/- 1.4) x 10(32) ergs s(-1) (0.1-2.0 keV), which is located very close to the cluster center. There is evidence for X-ray variability on a timescale of less than or similar to 1 day. The presence of this LLGCX in such an open globular cluster suggests that dense stellar systems with high interaction rates are not needed to form LLGCXs. We also searched for diffuse X-ray emission from NGC 288. Upper limits on the X-ray luminosities are L-X(h) < 9.5 x 10(32) ergs s(-1) (0.52-2.02 keV) and L-X(s) < 9.3 x 10(32) ergs s(-1) (0.11-0.41 keV), These imply upper limits to the diffuse X-ray-to-optical light ratios in NGC 288, which are lower than the values observed for X-ray faint early-type galaxies. This indicates that the soft X-ray emission in these galaxies is due either to a component that is not present in globular clusters (e.g., interstellar gas or a stellar component that is not found in low-metallicity Population II systems) or to a relatively small number of bright low-mass X-ray binaries.