I don't think "come down on the price" is grammatically correct. Though I will agree that in common speech you can hear it quite often.
The idiom "come down in" usually means "lessen".
"The man came down in price for the camera when he realized he was charging too much for the item."
"Bob hopes the winds will soon come down in severity, otherwise he will lose the roof of his house."
"His actions of late have resulted with people thinking he has come down in the world."
The idiom "come down on" means "scold", "cast judgement", "criticize".
The judge came down hard on the thief when he delivered his judgement. (scold)
The committee came down on the side of the neighbourhood association in the real estate dispute. (cast judgement)
John felt his mother came down too hard on him for such a trivial mistake. (criticize)
Finally to whether "come down in" or "come down on" is the correct phrase with respect to price, I googled "come down in price" and "come down on price"
You must wrap quotation marks around each phrase or Google will analyze each word, not the complete phrase.
The results are:
"come down in price" 405,000 hits
"come down on price" 75,200 hits
As you can see "come down in price" is the favoured construction without the use of "the" with price making it a general price.
Curiously the addition of "the" makes a HUGE difference.
"come down in the price" 4 hits
"come down on the price" 1,510,000 hits
When you add "the" making the word "price" a specific price, the sentiment overwhelmingly is in favour of "come down on the price".
I can only conclude the preference depends on whether or not you are considering a specific price or not. Grammatically though "come down in price" seems to be the favoured construction according the the definitions I have found.
