In the Sky Boat Song, there’s the line:
“That’s” looks wrong to me - I would have thought it should be “who’s”, since it’s a person who is born to be king.
Which is preferable?
It’s not ungrammatical – it’s a usage question. Who, whom, and whose are usually preferred over that and which as the relative pronouns referencing people – but not always. And the particulars vary with ‘tone’, level of formality, and local dialect far more than a typical grammar/usage book is prepared to allow.
I always thought the line was, “…who was born to be King.”
Call me a crank, but I don’t think which and that should be used to refer to people. Ever.
You’re a crank. You’re also wrong, with no support from any usage authority.
It’s true that in most cases, in formal writing, who is preferred to that when referencing persons. It’s a matter of style, though, and many instances exist in which that would be the stylistic choice.
For casual writing, especially for a song lyric, those formal rules are worse than useless. They just lead people, like you, down the wrong path.
It’s a gateway language rule!
It’s just style. Who’s and that’s are both fine.
MPSIMSy Aside only tangentially related to the grammar question: Back when Flora MacDonald (a descendant of old Sir John A.) was MP from Kingston, ON, and Minister of Silly Walks or some such, it happened that Prince Charles visited Ontario. And as a bit of amusing statecraft, the Minister took the Prince of Wales out on her yacht, to some island that stood in for Skye. Channel 11 made a big deal of it, of course. 
That’s what I was taught, by an English teacher who was very much a stickler for this sort of thing . . . though I realize these things are more flexible today than when I was in school.
“Which” is not generally used to refer directly to persons. It’s technically incorrect in formal usage, and I think most people would find it too weird-sounding for casual use too. “The girl which waits at the bus stop every morning…” 
:dubious: Nah.
But “that” is a recognized alternative to “who” or “whom”, and is often used, especially in place of “whom”:
How about the other way 'round?
Meet me by the lamppost whose light is out.
I do not understand why the words (which, that, who, etc.) are necessary at all. In the mentioned examples, could you not correctly say “Carry the lad born to be King” or “The Girl I Marry” and “You’re the One I Want”? Are they not inserted simply to make the words equal to the notes of the song?