There are various methods of treating mould or fungal attack on baskets which can be divided to a certain extent into prevention and cure. To prevent a fungal attack the basket must be dried and then painted with either varnish, oil or a wood preservative. A lot of baskets seem to be varnished these days but I find this treatment not to be aesthetically pleasing although there may be a case for varnishing fishing creels and other such baskets which are exposed to the weather. Wood preservative will probably work but might leave a residual smell on the basket. Treating with oil would seem to be the best option. Linseed oil is the usual choice but teak or perhaps cedar oil might be used. The oils must be diluted with white spirit in the ratio of about 2 parts spirit to 1 part oil. There may be problems with the absorption of the oil in baskets made of unstripped willow.
To fix baskets which already have a fungal attack you can plunge the basket in a container containing Oxalic Acid and Water and allow to dry. An alternative to this is to paint the basket with bleach (sodium hypochloride) and dry. Both these methods work and give immediate results but I am not sure that the effect is long lasting and baskets treated in this way can get further attacks. A more traditional way of treating mould was to place the basket (wet) into a container in which sulphur was burnt. The basket was usually left overnight in this container. This method is very good if you are using white willow.
In these days of Organic farming and food scares the treatment of baskets using any of these methods might be a bit suspect. Probably the best option is to avoid the situation completely by keeping the willow and baskets in a warm dry room.