One trick of the trade is to fill the bottom of the basket with filler. This will give the illusion of a fuller basket and make the gifts easier to see.
Overfill the basket, as it will be weighted down as gift items are added.
Begin adding the biggest and tallest items toward the middle or back. Aim for balance in weight and arrangement.
To hold some items in place, use tape to attach them to each other, to the basket handle, or to the container. The deep filling makes it easier to keep things still by "sinking" them in a bit.
Continue arranging gifts and then "fill out" the look of the basket. Make sure the paper or shred shows through. Fill in with more, if necessary. Add little items such as silk flowers or whatever works with the theme. Gift cards are a good small-space addition.
Wrap with cellophane or clear cellophane bags made for baskets - available at craft shops or dollar stores near the wrapping paper. If using a bag, gather at the top of the basket and tie with a ribbon. Leave some of the plastic fluffed out above the ribbon. Add a pretty bow or a matching trinket. Tape the bottom of the bag tight to the basket if it tends to stick out.
If preferred, simply add a bow and skip the covering.