How do I create a new MongoDB from the command line with auth turned on?

Beware of the fact that addUser function got deprecated since version 2.6.

Use db.createUser() and db.updateUser() instead of db.addUser() to add users to MongoDB.

Your code should be like below :

mongo -u username -p password admin --eval "db.getSiblingDB('new_database').createUser({ user : 'new_user', pwd :  'new_password', roles : [{ role: 'readWrite', db: 'new_database'}]});"

You can use db.getSiblingDB(name) to reference the new database and add a new account, eg:

mongo -u username -p password admin --eval "db.getSiblingDB('new_database').addUser('new_user', 'new_password');"

Now you should be able to authenticate from the command line using -u and -p:

mongo -u new_user -p new_password new_database