"Laundry baskets" can be a great synonym for clothes baskets as it refers to the same thing. "Hamper" is another great synonym as it not only refers to the basket but also to the task of collecting and storing dirty clothes. "Bin" is yet another synonym that can be used interchangeably with clothes baskets, as it refers to any kind of container used for storage. "Basketry" is another synonym, but it is more exclusively used for woven baskets made out of materials like wicker or bamboo. Overall, these words can prove to be very useful when describing or looking for alternative names for clothes baskets.