Sometimes, urban legends are formed around certain breeds of dogs and the Chihuahua is no exception.
Many people will tell you that Chihuahuas are "bad with children". The reality is, many children are bad with Chihuahuas and many Chihuahua owners do not take the time to properly train or socialize their dogs.
Chihuahuas as a breed are not naturally antagonistic toward children. With proper socialization and supervision, these bright little dogs can make very good pets in families with well-behaved children.
The best time to socialize and condition a dog is from the moment you bring him home. Puppies are especially trainable because they don't have bad experiences to overcome.
When you bring home a new Chihuahua, your children will be excited and want to gather around the dog. This is a mistake, especially for a small dog like a Chihuahua. Imagine how you would feel, transported into a world of giant, loud strangers, all wanting to get close to you.
Give the dog some space. Confine him to one puppy-proofed room and have everyone sit down and just let him explore and approach people when he's ready.2
Teach your children to be calm and gentle.
Shouting with excitement, grabbing the dog, and other frantic activity (common to young children) is a sure way to panic the dog and incite a bite.
Your children don't have to permanently tip-toe around the house and speak in only muted voices, but giving the Chihuahua a few days of calm and gentle behavior will teach him that none of you mean him any harm.3
Treat the Chihuahua like a big dog.
As the dog gets used to your household and the people in it, make sure you train the Chihuahua to obey the rules of the household - this means he needs housebreaking, needs to be taught what is and isn't okay to chew, needs to be reprimanded for mouthing or nipping anyone, and needs to learn basic obedience commands like stay and coming when called.
It is tempting to treat the Chihuahua as if he were truly a toy, an inanimate object, instead of a dog. This is always a mistake, and leads to a poorly socialized, untrained pest instead of a happy and mentally healthy member of the family.
Your children should be involved in the training process - even very young children can help teach a dog to sit for a treat.