Ginger beer is generally a non-alcoholic soda and I don't believe the recipe you linked to was meant to contain much alcohol either. I've made ginger beer like this recipe many times. Basically, you mix your ingredients, seal the bottle and leave it on the counter until the plastic no longer has any give when you squeeze it. Then you place it in the fridge to slow/halt the fermentation process. Once chilled, drink your finely carbonated beverage.
By loosely capping the bottle, the CO2 is obviously going to escape. If you let it go long enough, you'll have an extremely dry and flat alcoholic ginger flavored drink that probably won't be very pleasant.
EDIT: When I make a recipe like this I simmer the water, sugar, ginger mixture for about 20 minutes then chill to pitching temps. I strain this mixture before putting it in the bottle and adding yeast.