If you receive a bottle of wine in a wine bag or a gift in a gift bag, keep that bag and use it for your next gift-giving. Many of these bags are already made from recycled materials, which means you're being even more green.
Rather than grabbing for ribbon, try some colored raffia or hemp twine, which give the same look as yarn with a more green approach. Another option is silk sari yarn, which is again more green than typical yarn.
This type of tape is made from recycled Kraft paper instead of petroleum-based products. Adding fewer creases to your packaging will also mean even less of this product will be required.
Cornstarch packaging "peanuts" are just as effective as other alternatives and they dissolve in water with no toxic effects. You can even place these peanuts outside for birds to grab as part of their next nest-building process. If birds don't grab the peanuts, the rain will wash them away.
Wrap presents in the comics section of old newspapers, reuse old boxes instead of buying new gift boxes, and use old box packaging to safely secure items.
Using Tulle, semi-transparent fabrics and lace with gift baskets and inside gift bags will allow others to reuse those products for their own gift-giving. These products also tend to be more eco-friendly than aluminum and other hard material products.