"Drunk as a skunk” means to be highly inebriated. Skunks aren’t particularly known for drinking; the saying exists because of the rhyme. “Drunk as a skunk” has been cited in print since at least 1940.
Although skunk might refer to the aroma of alcohol etc, the rhyme seems to have motivated this idiom.
drunk as a skunk
1. (simile, colloquial) Highly inebriated.
This term follows the common pattern of omitting the first “as” of the full form “as drunk as a skunk”.
24 November 1940, San Diego (CA) Union, “Out of My Mind” by Katharine Brush, pg. C3, col. 6:
‘Drunk as a skunk but we got him registered in time.”
15 December 1942, Oakland (CA) Tribune, “Cohn-ing Tower” by Art Cohn, pg. 28, col. 1:
Except for rare few teetotalers—the late Howard-Jones, Double-A Stagg, Babe Hollingbcry and one other probi whose name escapes me—everyone got drunk as a skunk.
19 May 1947, The Oregonian (Portland, OR), “Behind the Mike” with William Moyes, pg. 13, col. 2:
A couple of hours later, he was drunk as a skunk, but at last reports still had the candy.
By Grace Metalíous
New York, NY: Messner
“He’s in the woodshed, drunk as a skunk,” said Ginny to those who came to inquire for him.
Google News Archive
6 August 1958, Daytona Beach (FL) Morning Journal, ‘Dear Abby” by Abigail Van Buren, pg. 4, cols. 4-5:
DEAR ABBY: My husband got drunk as a skunk and came home at four in the morning with lipstick smeared all over his face.
29 October 1966, Billboard magazine, pg. 105 ad:
But the real fun comes in watching an all-thumbs customer who has one small drink get rated “Drunk as a Skunk,” while a skillful (or lucky) heavy drinker rates “Sober as a Judge.”
(The Northwestern Corporation—ed.)
OCLC WorldCat record
Drink like a [fish], drunk as a [skunk], sink like a [rock], dead as a [duck]; safe [boat] ing, no [alcohol] : your safety, our concern.