A father bears the obligation to train his sons to perform the Misva of Sefirat Ha'omer once they reach the age of Hinuch (training in Misva observance), which is generally defined as approximately six or seven years of age. He should therefore recite the Beracha and count the Omer with his sons each night during Sefirat Ha'omer. Even if a child missed a day of counting, the father should nevertheless have him count on the remaining nights with a Beracha. Although an adult who misses a day of counting does not recite the Beracha when he counts on the subsequent nights, in the case of a child, he should nevertheless be taught to recite the Beracha as part of his training in the proper performance of this Misva.
An interesting Halachic irony arises in the case of a boy who becomes Bar-Misva during the Omer period. We follow the position of the Behag ("Ba'al Halachot Gedolot"), that the forty-nine days of counting constitute a single Misva, such that one missed day undermines the fulfillment of the Misva even with respect to the other days. For this reason, as mentioned, a person who misses a day of counting no longer counts the Omer with a Beracha. By the same token, Hacham Ovadia Yosef rules that if a child becomes Bar-Misva during the Omer period, on the day he turns thirteen he must discontinue reciting a Beracha before counting. During the previous days of the Omer, he counted as a minor, who is not, strictly speaking, obligated in Misvot, and he has therefore not performed an official "counting" of the Omer until his thirteenth birthday, when he becomes obligated. Hence, he cannot perform the complete forty-nine-day counting – just like a person who missed a day or several days of counting. Therefore, ironically enough, even though he had been counting with a Beracha before his Bar-Misva, upon reaching the age of Bar-Misva he may no longer recite a Beracha before counting the Omer.
Hacham Ben Sion Abba Shaul (Israel, 1924-1998) disagreed with this position, and held that a young man in this situation should continue counting the Omer with a Beracha. Nevertheless, in light of the fundamental rule of "Safek Berachot La'hakel" – we do not recite Berachot in situations of Halachic doubt – a boy in this situation should not recite the Beracha, in accordance with the ruling of Hacham Ovadia.
Summary: A father must train his children to count the Omer each night with a Beracha once they have reached the age of Misva training. If a child missed a day of counting, he should still count with a Beracha on the subsequent nights of the Omer. If a child becomes Bar-Misva during the Omer period, he should count without a Beracha from the day he becomes Bar-Misva until the end of the Omer period.