The vast majority of plants sold as baskets will do just fine in the full sun. Always check the tag though. Common plants to stay away from are New Guinea impatiens, fushia, coleus, ect. The petunia baskets, bacopa,bidens, calibrachoa,ipomea,onoethera,portulaca are all common and do vey well in full sun. The reason most baskets fail is simply water, or lack there of. Whichever basket you choose, remove it immediatly from the plastic 10" pot that it was grown in. Loosen the roots very well. ( I almost always come very near to bare rooting the annuals that I grow wether in a basket or in the ground. Replant the loosened root mass in an affordable coco or moss lined basket available just about anywhere at least two inches larger in diamater than the original.
repot with a good potting mix and water, water, water. There are many little gagdets around that can be mixed in with the soil to conserve water or little bottles that attach to the pot and they are pretty much worthless.
The important thing to remember is that you cannot, no matter how much you try, give that basket too much water! Soak the devil out of it in the morning and again in the early evening.