柿が赤くなると医者が青くなる
Redder persimmons, darker blue doctors.
Inverse relations between persimmons and doctors.
From 広辞苑:
柿が赤くなる秋は天候がよいので病気になる人が少なく、医者は商売にならずに青ざめる。秋の快適な気候をいう。
Reddening persimmons signify autumn, a season of good weather. As less people fall ill during such times, doctors lose business and thus turn a greater shade of 'blue'. This saying is used to describe the pleasantness of autumn weather.
From ことわざ学習室:
このことわざは、特に柿だけではなく、秋頃は、みかんや、ゆずなども、実をつけて、色づき、食べ物が豊富で、過ごしやすくなるので、病人が少なくなるということのようで、秋の過ごしやすさを言っているようですが、たぶん、昔は、冷蔵庫などがなかった為、夏の間の熱い時期は、食べ物が腐りやすく、食中毒なども多かったためではないかとも考えられます。実際、これらの食べ物には、ビタミンＣが多く含まれているようです。また、「橙が赤くなれば医者の顔が青くなる（だいだいがあかくなればいしゃのかおがあおくなる）」とも言うようです。
This saying does not specifically refer to persimmon trees.
Orange trees, citrus trees and the like bear fruits and change colours in autumn; as bounty is plentiful, the season is easier to pass and fewer people become sick. This saying is used to describe autumnal tranquility.
The saying perhaps makes more sense once compared: refrigerators did not exist in times past, so foods were quick to rot during hot summers, causing many to suffer from food poisoning. It is noted that (autumn) foods are rich with Vitamin C.
An equivalent saying also exists: "When citruses turns red, doctors turn blue".