As an adoptive mom - I'd love to also hear what we can do to encourage our children to search if they'd like to. And how we can be supportive of that process (help, sounding board, etc).
The best way that an adoptive mom can encourage their child to search if they decide someday that they want to is let them know that:
1) If they decide to search that you will support them;
2) Searching is entirely normal;
3) You understand their need to search;
4) You will still love them if they search;
5) You are strong and mature enough to handle the news of a search without bursting into tears and asking what you did wrong.
6) You will not feel a search is an indication that they do not love you or are ungrateful.
Some of these may surprise you, but I hear adoptees constantly worrying that a search might appear ungrateful to their adoptive parents. Many adoptees are fiercely protective of their adoptive parents and will not search if they perceive that it might be hurtful to their adoptive parents.
Even if a search is an experience that an adoptee wants desperately, without the blessings of their adoptive parents, they may forgo a search. Adoptive parents can encourage their children by discussions about their birth family and adoption.
Not all adoptive parents react as violently to a search as their children might believe that they will. Don't make your child stew over what you might think. Let them know early on that a search will be okay with you. Consider as your children are growing up talking about their birth family and the possiblity of reconnecting with them someday.
One school of thought is that if adoptees need to talk about birth family, they will. I do not believe it is that simple. Instead, I believe that adoptive parents may need to bring up the subject at times. For the adoptive parents to talk about the birth parents signals that it is okay for the adoptee to do so as well. Adoptees need to know that the subject is comfortable for the adoptive parents. Many adoptees assume the subject of birth parents is painful for them, and avoid the subject entirely to spare their adoptive parents.