Who categorises an aerodrome?
Localiser Green
The answer to your question can be read in JAR-OPS 1, AMC OPS 1.975 'Route and aerodrome competence qualification' (page 2-N-2, if you have a copy handy!).
This supports the JAR itself, which requires that 'an operator shall ensure that, prior to being assigned as commander ... the pilot has obtained adequate knowledge of the route to be flown and of the aerodromes (including alternates), facilities and procedures to be used.'
The AMC (Acceptable Means of Compliance, rather like the US Advisory Circular) states in paragraph 2, 'The Operations Manual should specify a method of categorisation of aerodromes and specify the requirements necessary for each of these categories.' It then goes on to describe what requirements should suit the categorisations A, B and C.
Thus, the responsibility for aerodrome categorisation rests with the operator (subject to the acceptance, in the UK, of the assigned Flight Operations Inspector) and should match the size and handling capabilities/difficulties of the aeroplanes intended to be used.