The term cusp is an old one and it was first used in astrology and later in other contexts:
1580s, in astrology, "first entrance of a house in the calculation of a nativity," from Latin cuspis "point, spear, pointed end, head," which is of unknown origin. Astronomical sense is from 1670s, "point or horn of a crescent." Anatomical sense of "a prominence on the crown of a tooth" is from 1839. (Etymonline)
but the metaphorical expression on the cusp of something meaning “be at the time when a situation or state is going to change” became curiously popular only a few decades ago, from the 1990s as shown by Google Books.
While it probably derives from the astrological use of cusp, I wonder what may have caused its popularity which, as shown, has been increasing steadily and consistently in recent years.