Either of these sentences could be correct. It really depends on what meaning you are looking for. Let me tell you what each sentence means.
The first sentence means that you used the binoculars to see the boy. If you had not had the binoculars you would not have been able to see him. But the point is that you are using the binoculars to look at him.
In the second sentence, the boy is using the binoculars and you see him without using binoculars yourself.
So when you ask for the correct answer, I'm not sure what the question is. Both sentences use correct grammar, but they have very different meanings.
As explained in the answer above, the two sentences given in the question have distinctly different meanings. Both the sentences are grammatically correct. However, to improve the understanding of how one sentence can have ambiguous meanings, and how we can improve the clarity of sentences. I am giving below a grammatically correct sentence, which could be interpreted to mean any of the two statements made in the question. We will then see how this sentence can be modified to remove the ambiguity in its interpretation.
Consider the following sentence:
I saw the boy using a binocular.
Here the phrase "using a binocular" may mean that I was able to see because I was using a binocular. Alternately, the phrase could describe that when I saw the boy, he was using a binocular. The ambiguity of the sentence can be removed by modifying the sentence as:
I saw the boy by using a binocular.
I saw the boy who was using a binocular.
The first of the two modifies sentence has the same meaning as sentence 1) of the question, and the second modified sentence has the same meaning as sentence 2) of the question.
I think these two questions are different and according to me the first one is the collect one.