If I play Janna as a reaction to a battlefield where my opponent is attacking, does their unit get exhausted?

No — Janna doesn't exhaust anything. Her ability only heals your units here and moves up to one enemy unit from here to its base.

That attacking unit is probably already exhausted anyway — using its Standard Move to attack costs exhausting it (Rule 144.2). Janna's move just relocates it; moving via an effect neither exhausts nor readies a unit. The real effect of playing Janna there is that the attacker leaves the battlefield, so the attack fizzles — with no attackers present, the combat resolves without a Combat Damage step and you won't take damage from it.