Lowell Fulson wrote and recorded "Sinner's Prayer" in 1950, although the most famous version was recorded by former Fulson sideman Ray Charles in 1957. Until preparing this post, I didn't realize that "sinners prayer" has a specific--and controversial-- meaning for Baptists, evangelicals, fundamentalists, Pentecostals, and charismatics. It is not a term used by Catholics and mainline Protestants.
Here is what wikipedia says
any prayer of repentance, spoken or read by individuals who feel convicted of the presence of sin in their life and desire to form or renew a personal relationship with God through his son Jesus Christ. It is not intended as liturgical like a creed or a confiteor. It is intended to be an act of initial conversion to Christianity, and a way to receive salvation. It also may be prayed as an act of re-commitment for those who are already believers in the faith. Often, at the end of a worship service, an evangelist will invite those desiring to receive Christ (thus becoming Christians) to repeat with him the words of some form of a sinner's prayer. It also is frequently found on printed gospel tracts, urging people to “repeat these words from the bottom of your heart”Regardless of whether Fulson had church experiences that lead to writing this tune, it is a classic.