Should I ever use a normal attack?

There is no reason to ever use a standard attack as a Barbarian, or any other character. The design philosophy that the team was aiming for was to allow people to use skills as much as possible, as that is the most satisfying way of taking out the baddies. Obviously this is somewhat limited by the resource systems set in place, but since the Barbarian, Monk and Demon Hunter all have resource generating skills, and the Witch Doctor and Wizard both have low to zero cost skills, it's obvious that Blizzard wanted people to use skills as their primary attacks, and not just swinging a weapon.

However, there is nothing stopping you from using a standard attack. Given that we've not seen content past Act 1 normal, there may be situations where killing multiple enemies at once is a bad choice, and if your default attack is bound to an AOE ability, (e.g. Cleave for the Barbarian, Crippling Wave for the Monk, etc.) you may be left to using a default attack.

There is at least ONE REASON to use a normal attack. There is an achievement called "Bashinishu". You must damage Bashiok with Rakinishus blade which has very low damage.

For example, take me, a wizard: I want this achievement, therefore I don't want to kill Bashiok with magic before I damage it with this sword.

I'm actually trying to do this right now.... I know before you could just drag skills from your hotbar to un-equip them and use a normal attack, but it's not working now... :(

This post is years late but it could be used to break a door open or barrier if you have a build with attack skills that all have cooldowns and can't wait 5 seconds for a cooldowns. Example hell tooth witch doctor.