I think this is a very old phrase that dates back to the Anglo French wars.
At that time it was law for every man in England to practice archery on the village green on Sundays.
'Fin' refers to the flights, feathers, vane or fletch of an arrow.
During practice a man was nominated to retrieve the shot arrows from the targets and to avoid being hit by a stray arrow, he would shout 'Fin it down', meaning to point the arrows down to the ground, while he collects the arrows.
So basically it means, be careful or get it right