The Grammy Awards first included a separate rap category in 1989, when Best Rap Performance was introduced.
The inaugural award was presented at the 31st Annual Grammy Awards. D.J. Jazzy Jeff and the Fresh Prince won for “Parents Just Don’t Understand,” defeating nominees including J.J. Fad, Kool Moe Dee, LL Cool J, and Salt-N-Pepa. The new category marked the Recording Academy’s formal recognition of rap within the Grammy system.
Some references create confusion by discussing the music released in 1988, because the first nominees’ recordings came from that eligibility period. The ceremony itself took place in 1989, so 1989 is the standard answer to this question. Rap categories later expanded: Best Rap Album arrived in 1996, Best Rap Song in 2004, and additional performance categories followed as the genre developed.