All of the characters in Hemingway's short story have their “opiates”, the things that dull the pain of their daily existence. Cayetano has his gambling, Sister Cecilia her religion, and Mr. Frazer a radio.
Mr. Frazer has the radio on whenever he can get a reception. He plays it all night long, but so low that it can barely be heard. There are a number of reasons for this. First and foremost, Mr. Frazer suffers from a nervous condition, and having the radio's volume turned up too high would undoubtedly fray his already damaged nerves. Secondly, the radio needs to have its volume low at night so that the other patients in the hospital are not disturbed.
However, the significance of the low radio volume is much greater than this. As we've already seen, the radio has become Mr. Frazer's opiate, something that soothes his pain and suffering. The very fact that the radio is there is enough to prevent Mr. Frazer from suffering too much. It's the presence of the radio that matters, not the volume at which it's being played. In that sense, one could say that Mr. Frazer only needs a relatively small dose of “opiate” to help soothe his pain.