The best replacement for baskets is boxes.

The best alternatives to baskets are rated as following:
★★★★★  boxes
★★★★  containers
★★★★  buckets
★★★★  alba berlin
★★★★  bins
★★★  boxes for hampers
★★★  flowers
★  towers
★  bowls
★  hampers
★  longaberger
  artland dragons
  bouquets
  bins storage
  phoenix hagen

Your alternative for baskets is not listed? Suggest!
Speaking about baskets, one probably shouldn't forget about lids for storage.

Other search terms related to baskets and their ratings are:
★★★★★ lids for storage
★★ shelves
★★ drawers
You have in mind a companion for baskets and it is not yet here? Propose!



Google Trends

Scale is based on the average worldwide traffic of baskets in all years. Click on graph for more.
baskets
boxes
containers