The most important thing to do is be more selective about the wine you purchase. Natural cork is a way better option than synthetic corks and metal screw tops, which don't biodegrade and are harder to recycle. Next time you're shopping for wine, look for the "Real Cork Inside" acorn on the label.
Once you've finished drinking your wine, hang onto the corks instead of tossing them into the trash. There are plenty of options for recycling them.
Here's a video of a wonderful TEDx talk given by Patrick Spencer, executive director of the Cork Forest Conservation Alliance (CFCA). He speaks about why it's important to maintain natural cork forests.
The CFCA has created the Cork ReHarvest campaign and partnered with a variety of companies to recycle used wine corks and turn them into a variety of products. You can drop your corks off at any Whole Foods location. If you don't live near a WF, here is a list of collection sites.
ReCORK is another company that recycles used wine corks. The cork they collect is used to manufacture shoes for the SOLE brand. Find a cork collection site near you using their location finder.
You can also ship your used wine corks to Yemm & Hart, a materials manufacturer that actually pays 1/2 cent per cork. The minimum package weight requirement is 15 lbs, so this option is probably best for groups of wine drinkers or businesses that go through large quantities of wine. But maybe you can package up 15 lbs of wine corks on your own. No judgement!
You can even compost your natural corks. As I mentioned, they won't biodegrade quickly, so you may want to cut them up into small pieces before you toss them into your compost pile. Try giving your corks a quick whir in the food processor to break them down. Having bits of cork in the soil will actually help to retain water for your plants!
And if you're crafty, there is no shortage of cork craft ideas on Pinterest!
If you want to support cork recycling even further, you can purchase products that are made out of used wine corks. When you purchase one of these products, you are supporting the maintenance of natural cork forests and keeping wine corks out of the landfill.