How can I keep red or white wine from spoiling once it has been opened?

Keep the wine away from oxygen. When wine oxidizes, it degrades quickly and soon converts to vinegar. 
 
If you wish to prevent oxidation, keep a few empty half-bottles on hand, fill them to top with the leftover wine, and then cap the bottle with a cork or plug so that some of the wine spills out. This will prevent an air bubble from occupying portions of the bottle. Another alternative is to buy a Vac-U-Vin®, which removes oxygen and creates a vacuum to preserve the remaining wine. Yet another: Many wine shops market small cans of inert gas that replace oxygen when the gas is sprayed into the bottle.
 
A bottle that has been opened will keep for several days -- even up to a few weeks -- if it’s refrigerated and oxygen-free.
 
Source(s):
Wines.com "Can Leftover Wine Be Saved?"
Wineinsiders.com "Here Are Our 10 Most Frequently Asked Questions"
Meals for You: "Kitchen Tips : Wine Shelf Life"