Actually, both expressions are considered correct, but “on the one hand … on the other hand …” is preferred by many dictionaries while “on one hand … on the other hand” is listed as an alternative form. However, intuitively the first “the” seems illogical, because you are just referring to one of your hands. You wouldn’t say, for example, “I wore a glove on the one hand and nothing on the other”, unless you were waving one of your hands while saying so, and you wanted to underline the fact that this had been the hand with a glove.
I’ve read an explanation the first “the” originates really in gesticulation using one hand while telling the phrase. I was interested whether other languages follow the same pattern, so here comes a little table:
|German||Auf der einen Seite … auf der anderen seite||On the one side … on the other side||This expression copies the structure of the much less used English expression “on one side … on the other side”, but, unlike English, it uses the definite article, as in “on the one hand …”.|
|French||D’un côté … d’un autre côté …||At one/a side … at another side||Interestingly, French uses indefinite articles in both parts of the expression.|
|Spanish||Por una parte … por otra parte …||For one/a part … for other part||Spanish uses an indefinite article in the first part and no article in the second part.|
|Czech||Na jednu stranu … na druhou stranu …||On one side … on the other side||Although there are no articles in the Czech language, the feeling is that of the first “side” being indefinite whereas the second one being “the other one”.|
Overall, it seems that the usage of articles is not governed by any deeper logic; one just has to get used to the usual expressions in the respective languages.