The Singapore-based firm Asia Pacific Breweries (http://www.apb.com.sg/) have a brewery on Hainan and their Anchor beer is common all over the island. As anywhere is China, a range of both Chinese and imported beers are widely available as well. Hainan also has a number of locally-brewed pineapple-based beers, odd but worth a try.
See the drink section of the China article for information on other booze; Hainan is much like any other province for this.
The island is covered with coconut trees. So drinking fresh coconut milk is a must!
