Well, first, we have to answer what a juvenile delinquent is and then the history of criminal responsibility in the US. Figuring out what a juvenile delinquent is outside of the legal system is not an easy thing to do- is it just breaking the laws of the country? Is it breaking the unspoken social laws and norms, as well as legal violations? If so, there have been juvenile delinquents throughout history. There were youth subcultures probably since before Roman times, and there were always youth offenders- from pickpockets to rowdy apprentices rioting in Medieval London!

A juvenile delinquent is, legally, someone under the age of 17 who commits an act that would be charged as a crime if they were an adult, and juvenile delinquency is dealt with in juvenile courts or the criminal justice system. The phrase 'juvenile delinquent' has long been a part of the American conscious, for example, this poster in 1936. The first reference to 'juvenile delinquent' that I can find is in 1817 (that I can find, where the phrase was mentioned in the Observer), see here, being mentioned infrequently before not being mentioned often between 1859 and 1903, when it suddenly experienced a massive upswing in usage, although the phrase was most commonly used in the 1950s and the 1970s. However the first use of the phrase 'juvenile delinquency' is in 1805, much earlier, showing that identifying those who were involved in juvenile delinquency by that took much longer.

A juvenile delinquent can be identified by many similar terms, such as juvenile offender, young offender, youth offender and so on. The earliest use of one of these phrases in writing, that I can find, is from 1680, where, in the Juvenile Rambles of Thomas Dangerfield, it is stared that 'young offenders incur the displeasure of the Law, ere they hardly understand what the Law forbids.'

The idea of a young offender is therefore an old one, but the phrase juvenile delinquent must only be two centuries old at most. It sounds official and grand, and that was probably why it was used- for reformatory schools, organisations and newspapers, being able to use 'juvenile delinquency' in their names, titles or articles lent a scientific, intelligent air. It probably migrated to America naturally, with British people emigrating to the US and bringing the phrase with them, and spreading by Americans 'copying' and following the trend of calling young offenders, juvenile delinquents.

This might be a useful resource for the history of the development of the youth justice system- it's for the UK, not the US unfortunately, but it shows the development of how people viewed 'juvenile delinquents'- and the developments in the treatment of young offenders too.