J2 dependent visa dating over 20 years ago

It should be safe to assume that he had a J-2 visa if he went with his father and his father had a J-1 visa.
I don't think there's a way to get those records from the US government. Maybe the original J-1 sponsor could provide the J-1 start date or employment start date, but this is a long shot.
For something that happened over 20 years ago, I think an estimated date would be acceptable if there is no way to find the exact date. Indicate that the date is estimated and the reason why the exact date is unavailable---for example, the old documents were lost, or he no longer has access to them.

Source: stackoverflow.com