Escort was originally a military and masculine term:
1570s, in military sense, from Middle French escorte (16c.), from Italian scorta.
which was used figuratively from the first half of the 20th century:
The sense of "person accompanying another to a social occasion" is 1936.
From the latter sense, probably, the more recent connotation of prostitute:
c. A person, often a prostitute, who is hired to spend time with another as a companion. (AHD)
(American English) a prostitute, especially one who goes to social events or on trips with the person who pays them. (Longman Dict.)
It is not clear when this euphemism for prostitute caught on. In Italy, for instance, the usage of the term escort became common just a couple of decades ago, when Berlusconi was in power, because of his many “affairs” with girls.
Following a few comments suggesting that the term escort is not commonly used in the context of prostitution, I add this related link:
- An Escort agency is a company that provides escorts for clients, usually for sexual services.
When did the term escort start to be used to refer to prostitutes?
Was it originally an AmE usage?