It is idiomatically preferable, in my view, to include the in both parts of the correlative expression On the one hand/on the other hand. I know that many competent writers omit the first the, so that the expression reads On one hand/on the other hand. And I understand the argument in favor of omitting it: after all, we do not normally insert the in comparable expressions. We would not typically say, in describing the contents of two grocery bags, In the one bag/in the other bag.(Or maybe we would; I certainly would not object to that expression.) I also acknowledge that On one hand may be at least as popular (in American English, though not in British English) as On the one hand.
But I nonetheless believe, for several reasons, that On the one hand is the better choice:
1. It has history on its side. On the one hand is the traditional idiom. It sounds familiar and natural. Why omit the when it is doing no harm?
2. Authoritative dictionaries treat On the one hand as the standard expression, meaning “from the first perspective” or “from one point of view.” Look it up (under the entry for hand).
3. There is a comforting balance and rhythm to the expression that is sacrificed when the is omitted.
4. The expression is figurative, not literal. We’re not talking about actual hands the way we might about actual grocery bags, so resort to comparable expressions does not carry much weight.
5. On the one hand is likely to be less noticeable, less jarring, than On one hand. Every time I see On one hand, I feel the need to insert the after on (just as I cannot keep myself from adding a missing of after couple in the expression couple of). Why risk calling needless negative attention to your writing?