Several recent studies with the ROSAT and ASCA satellites have reported X-ray emission from Class I protostars. We present here an X-ray survey of 156 protostellar sources using archived ROSAT PSPC pointed observations. Targets are drawn from four samples: low-temperature objects in nearby star forming regions, Herbig-Haro outflow sources, CO outflow sources, and Class 0 embedded protostars. Over 90% are not detected at the 3 sigma limit over the 1.0-2.4 keV band, with upper limits of 0.1-40 counts ks(-1) corresponding to log L-X similar or equal to 28-33 ergs s(-1). Of the 11 detected objects, 10 are either Class II or III stars or previously detected Class I protostars. One new Class I source, Ced 110 IRS 6 in the Chamaeleon I cloud, is found. The X-ray luminosity function of protostars is calculated, but is quite uncertain. We conclude that X-ray emission from protostars has a broad luminosity function and that very X-ray luminous (L-X greater than or similar to 10(31) ergs s(-1)) protostars are rare.