Although the two spellings are often seen used interchangeably, recommended modern usage is to distinguish between them:
stanch: verb. to stop the flow of something, usually blood.
staunch: adjective. (of persons) strong, standing firm and true to one’s principles.
Pronunciation differs among English speakers, but /aw/ is the most commonly heard vowel sound in staunch. The OED gives /ah/ as the first pronunciation for stanch, but Merriam-Webster gives /aw/ first for both staunch and stanch. OED gives two alternate pronunciations for each word. M-W lists five alternate pronunciations for each.
Stanch occurs as an adjective in old books and periodicals (1930 and earlier), but modern usage favors reserving stanch for use as a verb and staunch as an adjective.
H.W. Fowler, Modern English Usage
stanch, staunch. The adjective is usually staunch, the verb stanch.
Chicago Manual of Style
staunch; stanch. Staunch is an adjective meaning “ardent and faithful”
Stanch is a verb meaning “to stop the flow”; it is almost always used in
regard to bleeding, literally and metaphorically
stanch, staunch. Stanch is a verb. Staunch is an adjective.
However, not all journalists consult Fowler, CMOS, or AP:
Gillard’s challenge is to staunch the bleeding (The Australian)
…the operators struggled to quickly staunch the flow of oil into the Gulf of Mexico. (Reuters article)
One should give Kennedy his due as a stanch fighter for what he believed in, (political website called American Power)
the senator … is a stanch supporter of the right to hunt and the right to bear arms. (USAToday)
Here are some examples from writers who do observe the distinction:
…he was a staunch friend of Poland’s Jewish community.
Staunch anti-Castro U.S. congressman to retire
Hospitality, government jobs help stanch D.C. unemployment tide
part of the Sixth SS Panzer Armee embarking for the Eastern Front to try to stanch the Russian advance.
Does it matter? At least one recent grammar test, the Dow Jones Grammar Test 2009, includes an item that requires a choice between stanch and staunch. (You must enter an email address in order to access the test.)