Option A: He didn't and is telling a lie or he has a false memory.
Option B: He did and his arrest didn't result in a charge or conviction, or it was swept under the rug thanks to his father's money, power, and influence.
Knowing how rich families operate and knowing Biden was planning on a political career early on in his life, Option B seems the most likely to me.
You can be a racist and still be arrested at a civil rights protest, you don't have to be there to support a protest. From there political spin can make a rainy day seem like a day in the sun.
Also a man will find himself doing things he never would have done because a girl is involved. Liberal cuties have beguiled many a conservative man, for some strange reason liberal women are often drawn to conservative men.