According to Urban Legends Reference Pages: Television (Unthemely Behavior), the Star Trek theme song did have lyrics:
Beyond
The rim of the star-light
My love
Is wand’ring in star-flight
I know
He’ll find in star-clustered reaches
Love,
Strange love a star woman teaches.
I know
His journey ends never
His star trek
Will go on forever.
But tell him
While he wanders his starry sea
Remember, remember me.
Why did it have lyrics? It’s complicated, but basically because making up lyrics got Roddenberry half the rights to the theme’s royalties — and that was a lot of money.