Some dictionaries accept stanch and staunch as variant spellings of each other. But if you want to avoid confusion, use stanch as the verb meaning to stop the flow of, check, allay; and use staunch as the adjective meaning firm and steadfast or having a strong constitution. This is how the words are usually treated in edited books and news sources.
The push for P.E. is part of the effort to stanch the epidemic of childhood obesity. [Daily Press]
Santorum is a favorite of conservatives for his staunch opposition to abortion rights, which the Roman Catholic church also opposes. [Associated Press]
But some conservatives said they considered the protest a misguided attempt to stanch the bleeding from the November elections. [New York Times]
Since the duo revealed its plight a year ago they have found staunch support among parishioners and from the international Bronte fan club. [Guardian]