I think the greatest creator of empathy is breadth and depth of experience.
There is a very real sense in which we each only have our own perceptions and experiences to go on when we try to model what someone else is thinking - wether we do the modeling intuitively or we do it in a more conscious fashion makes no difference in this respect.
Thus the tendency of modern western societies to protect children from experiences that are damaging or dangerous is a real barrier to empathy with those from harsher environments.
It seems to me that it is important to give all people a wide range of experiences, some of which are very painful (though not lethal) so that it is possible for them to have enough experience to build empathy with others.
There seem to be to be a raft of other considerations, and two in particular are of critical importance.
We need to find mechanisms that are effective at getting individuals to experience the essential commonality of all human beings.
Our minds are essentially difference engines, they highlight for us the different, the unexpected, and the potentially dangerous. In the context of our evolution as edible primates in a world full of predators this was a very useful function. In today's context of a world full of people, with few predators, it is not powerful.
We need all people to know that every person is far more alike every other person than we are different, by a factor of at least 9:1 in the case of very different people, and 99:1 in most cases.
We all have the same issues. At some level we all lack self confidence, we all feel alone, we all feel different; however competent and assured we may seem in certain situations.
We are all, far more alike than we are different.
The other thing that would serve us all well if taught early relates to distinctions.
We all start from a very limited base. Our brains and bodies develop from a single cell to baby in 9 months, and within 5 years we are walking, talking, highly skilled little people, with a lot of cultural abilities.
Part of the process of learning about everything from nothing is making distinctions.
All distinctions start simple. The simplest distinction is 2. Either thing or not thing, light or dark, hot or cold, good or bad, tall or short. up or down, etc.
As we gain experience, we learn that most distinctions are made of much more than two. Light and dark become shades of gray, and then we add colours. Eventually if we come to understand the idea of infinity we see that most things are part of an infinite spectrum, and that there are many more than 2 states.
Unfortunately, some people do not easily give up their simple 2 state distinctions around good/bad and right/wrong - and there are whole institutions devoted to maintaining those simple distinctions at some level. Thus many of us tend to classify people into one of two (or a small number) of camps, based upon very limited information.
When we each get to see that process operating within us, it opens the door for a degree of empathy with others at different stages of personal development.
I love the Paulo Coelho quote from "Warrior of Light":
"Every Warrior of the Light has felt afraid of going into battle.
Every Warrior of the Light has, at some time in the past, lied or betrayed someone.
Every Warrior of the Light has trodden a path that was not his.
Every Warrior of the Light has suffered for the most trivial of reasons.
Every Warrior of the Light has, at least once, believed he was not a Warrior of the Light.
Every Warrior of the Light has failed in his spiritual duties.
Every Warrior of the Light has said 'yes' when he wanted to say 'no.'
Every Warrior of the Light has hurt someone he loved.
That is why he is a Warrior of the Light, because he has been through all this and yet has never lost hope of being better than he is."