For the baskets one possible solution is electro luminescent wire.
The wire is bright and is lit by battery packs of various sizes or direct current. Various EL Wire (The common term) and blinky items can be found on Michaels Lights http://www.michaelslights.com/products.htm
For a permanent installation it could be easy to attach the wire to the basket attaching the battery pack for night play.
Looking at the site there is a section with rope LED lighting that is solar charged. Fun fun.