The big fabric part, or envelope, isa usually made of rip-stop nylon. Polyester and other fabrics are sometimes used. The lower portions around the opening are usually made from a fire resistant material like Nomex, similar to what race car drivers and firemen wear.
Baskets are usually woven out of wicker since it is strong, lightweight and flexible. It is also very easy to maintain and holds up well under the stress of repeated landings.