Healthy kids for the most part are dependant on their parent's ability to lead by example and provide a healthy home life with a variety of situations. It's through these varied life experiences that children learn to develop the self confidence, independence and interdependence on others outside their direct family.
There are many causes of separation anxiety in children. I won't say that anything in regards to children is easy or can be broken down into a set of rules that work with every child. Each is different and as a parent of 8 with 4 adopted at various ages, our family knows first hand that what works for one child doesn't necessarily work for another.
That said, one thing what has served our family well is consistency whenever possible. Children are not born with a sense of major attachment but more with a sense of wanting to be secure. You can see readily see this in babies. While many new parents may not want to be away from their newborn babies, it's important to realize even they need to begin to develop ways to interact with others. It's only through the experience of being with other people can be skills be learned. Once again, parents can go a long way in developing mentally healthy infants and children if only by their attitude in a given circumstance.
With my children, it was important to make certain that they couldn't manipulate the situation when my husband and I decided to have a night out. All fits of crying and acting out were to no avail. It was only after they recognized that their manipulations didn't work did they begin to develop the self confidence and sense of independence that is critical to beating this form of Separation Anxiety Disorder. The key, at least for us was NOT to allow the child control the situation.
Another thing we were certain to do is follow through on our plans. When we said we would be back soon, we were soon. There was no attempt to manipulate our children and promise them gifts are other benefits to stay at a babysitters or relatives home.
Although separation anxiety is natural and the feeling of insecurity may be new, it's up to parents to help their children overcome the problem BEFORE they need counseling. There are resources available to help children overcome separation anxiety but a better approach is for parents to help their children develop the confidence and interpersonal skills necessary for life.
While not a professional child counselor, I can say that although my kids have had separation anxiety challenges, all have been able to overcome it. It's thorough parental consistency that children learn that they can trust what their parents say and do. This allows them to grow in confidence and independence.