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Battle Master fighters have the Sweeping Attack maneuver (PHB, p. 74), which lets the user use a superiority die to hit another nearby target. If the original attack roll would have hit the second target then it takes damage from the superiority die.

It's unclear whether the attack roll is used or just the number.

Would the damage dice being doubled from a crit apply to the 2nd target as well?

The second hit doesn't benefit* from the Critical Hit.

Let's look at the wording of Sweeping Attack:

When you hit a creature with a melee weapon attack, you can expend one superiority die to attempt to damage another creature with the same attack. Choose another creature within 5 feet of the original target and within your reach. If the original attack roll would hit the second creature, it takes damage equal to the number you roll on your superiority die. The damage is of the same type dealt by the original attack.

My emphasis in bold. You're not making an attack against the second target, you're checking to see if you're able to deal an amount of damage to it. As such, it can't benefit from the effects of a Critical Hit.

*Well, that's not entirely true. If you rolled a Critical Hit against your first target, then it would certainly hit the second target, regardless of AC, so the "benefit" of guaranteeing the attack connects is still there.