"Bring 'Em Down" is a neo-chantey first popularized by A.L. Lloyd & Co. (on the A SAILOR'S GARLAND album). The DT notes, "Reportedly composed by Bert Lloyd." According to this page, the liner notes from the Young Tradition's album (on which their rendition of the song appears) state:

A.L. Lloyd says that it was brought to Liverpool from the West Indies, where a variant of it had been a stick-fight song.

I wanted to add a reference that I stumbled upon to what seems to be the same song, in a traditional context.

Jekyll's book of Jamaican traditional songs, JAMAICAN SONG AND STORY (1907) has, categorized under the heading of digging songs, "Bring dem come" (pp184-185). The melody is clearly related to "Bring 'Em Down." Lyrics are as follows:

//
A Somerset me barn
bring dem come
bring dem make me batter dem
bring dem come
me would take me picker batter dem
bring dem come
A Woburn Lawn me barn
bring dem come
I will like to see dem batter me
bring dem come
A Goatridge me barn
bring dem come
I want to see dem jostle me
bring dem come
//

Notes to the song say:

"This is a digging contest. The Somerset men challenge their neighbours..."

So it appears that Lloyd did not write the song after all -- though the stereotypical chantey verses ("around the Horn we must go" etc) may have been his addition. Notably, his "In Liverpool I was born" exactly matches the style of the opening phrase here.

For people challenged with Jamaican grammar :) "Bring dem come" basically means "bring them on" or "let them come (and show us what they've got)".