story identification - Novel set in the future, children cannot change the class they are born into, one class is made uneducated by associating books with pain

The world is a world where children are born into a class, and they cannot change it. For example, some children are radiated in the womb, so that they do not want to move to a cold climate, so that they can be permanently employed as cotton pickers.

Then there are toddlers who are electrocuted (or physically punished) when they crawl to books, so that they associate books with pain, keeping them uneducated.

I believe the main character is a guy who wants to run away with a girl, looking for simpler life, as they are born in the high class.

Answer

"Babies having to associate books and electrocution" is most certainly Brave New World (1932) by Aldous Huxley. You can read the whole text here (I'm assuming huxley.net is a legit source).

The infants were unloaded.

"Now turn them so that they can see the flowers and books." [...]

The Director rubbed his hands. "Excellent!" he said. "It might almost have been done on purpose."

The swiftest crawlers were already at their goal. Small hands reached out uncertainly, touched, grasped, unpetaling the transfigured roses, crumpling the illuminated pages of the books. The Director waited until all were happily busy. Then, "Watch carefully," he said. And, lifting his hand, he gave the signal.

The Head Nurse, who was standing by a switchboard at the other end of the room, pressed down a little lever. There was a violent explosion. Shriller and ever shriller, a siren shrieked. Alarm bells maddeningly sounded. The children started, screamed; their faces were distorted with terror.

"And now," the Director shouted (for the noise was deafening), "now we proceed to rub in the lesson with a mild electric shock."

He waved his hand again, and the Head Nurse pressed a second lever. The screaming of the babies suddenly changed its tone. There was something desperate, almost insane, about the sharp spasmodic yelps to which they now gave utterance. Their little bodies twitched and stiffened; their limbs moved jerkily as if to the tug of unseen wires.

"We can electrify that whole strip of floor," bawled the Director in explanation. "But that's enough," he signalled to the nurse. [...]

"They'll grow up with what the psychologists used to call an 'instinctive' hatred of books and flowers. Reflexes unalterably conditioned. They'll be safe from books and botany all their lives." The Director turned to his nurses. "Take them away again."

The main character is from a high class indeed, and exits the "utopian" city he lives in to visit a reserve, where people live outside of this system. They're​ considered to be primitives. The sight of the reserve and the people who live in it frightens his girlfriend.

Here's the bit about making people hate cold. It's not the wombs that are radiated, it's the bottles in which the embryos are grown:

"Heat conditioning," said Mr. Foster.

Hot tunnels alternated with cool tunnels. Coolness was wedded to discomfort in the form of hard X-rays. By the time they were decanted the embryos had a horror of cold. They were predestined to emigrate to the tropics, to be miner and acetate silk spinners and steel workers. Later on their minds would be made to endorse the judgment of their bodies. "We condition them to thrive on heat," concluded Mr. Foster. "Our colleagues upstairs will teach them to love it."