This problem (sticky handles) is caused by a quirk of the chemistry of polyurethane. In brief it reaches a point through ageing where it will break down and become sticky to the touch. It's sometimes quite spontaneous after many years and can be accelerated by exposure to UV light.
The solution is actually quite easy. Disassemble the tripod handle so far as is possible, to remove the part affected and submerge in a solution of one teaspoon baking soda and one pint of cold water. Leave for 45 mins then rinse in cold water and dry. Voila, only the sticky stuff has dissolved in the alkaline solution, leaving you a clean fresh plastic surface beneath.