Aside from the song
Jessie's Girl I've only ever seen boy Jesses and girl Jessies. Personally if I had to use one I'd use
Jessie on a girl, but
Jesse for boys is definitely still around.
Jessie is almost always short for
Jessica and the
Jessica craze is over.
I know
Jesse comes from an old Hebrew name but it sounds extremely modern to me. I picture some kind of athlete / sporty person.