The story takes place in September, after the hot tourist seasons of Summer and before the cold rainy seasons of Winter. Santiago and Manolin discuss the current U.S. baseball season (roughly April - October) which could place the story anywhere in that period. However the month is named specifically several times:
"Keep warm old man," the boy said. "Remember we are in September."
"The month when the great fish come," the old man said. "Anyone can be a fisherman in May."
...high above were the thin feathers of the cirrus against the high September sky.
It was dark now as it becomes dark quickly after the sun sets in September.
(Hemingway, The Old Man and the Sea, Google Books)
There are two main reasons for this setting. The first is mentioned by Santiago; the biggest and best fish come in the later months, feeding before they seek warmer waters; alternatively, the bigger fish take advantage of smaller fish migration. Because of the large size of these Fall fish, they are harder to catch; "Anyone can be a fisherman in May" because the fish are smaller and more readily available, while the large fish of Fall are stronger and harder to catch.
The other reason for setting the story in September is to make Santiago's ordeal more difficult; the water and air are cold, with long nights and short days. This helps to sap Santiago's strength, although he continues to fight long after most would have given up. Through his long experience, Santiago knows how to survive on the ocean even in colder weather.
One last point is the contrast between Santiago's current life -- fighting large fish on a cold ocean -- and his younger life, when he traveled to Africa and watched young lions playing on the beach. The imagery for his dreams is warm and comforting, while the reality of the September ocean is cold and harsh.