Reducing impairing effects
Is not possible preventing fear effects (dodge is does not work with status effects) but there are few possibilities to reduce the duration of its effect.
As Monk you have only one option to be more resistant to fear, but you need to reach level 50 first.
Mantra of Evasion with Perseverance rune, reduces duration of all control impairing effect by 20%.
As Barbarian unfortunately you have no skill that can help you resist fear effects. Other 3 classes, Demon Hunter, Witch Doctor and Wizard are almost ranged, therefore they should be less affected by this threat.
It should be possible also use items that reduce the duration of control impairing effects, like this one (disclaimer: do not look at the red circle; this screenshot was used in another answer and I attached here only as example)
The best defense is a good offense
Finally, in my experience, having a lot of crowd control (stun, fear, etc.) help dealing with nightmarish mobs because when they are impaired they not trigger fear to you.