Appeal to a non-drinker that's holding dray up (7)
I believe the answer is:
attract
'appeal to' is the definition.
(synonyms)
'a non-drinker that's holding dray up' is the wordplay.
'non drinker' becomes 'tt' (Teetotaller abbreviation).
'that's holding' indicates putting letters inside.
'dray' becomes 'cart' (I've seen this in another clue).
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'tt' enclosing 'cart' is 'tcartt'.
'tcartt' reversed gives 'ttract'.
'a'+'ttract'='ATTRACT'
(Other definitions for attract that I've seen before include "To be liable to" , "Draw towards like a magnet" , "Exercise a magnetic influence" , "Bring in" , "Exert a pull upon" .)