Who said he was going to kill Baasha's family (and posterity) and feed them to the dogs and birds?

God

I [God] will take away the posterity of Baasha, and the posterity of his house; and will make thy house like the house of Jeroboam the son of Nebat.

Him that dieth of Baasha in the city shall the dogs eat; and him that dieth of his in the fields shall the fowls of the air eat.1 Kings 16:3-4