First Contact with an extraterrestrial civilization could happen in a number of ways. I think that there are three fundamental elements for a successful First Contact that would work under most circumstances: Transparency, global involvement and a careful process moving forward.
The SETI protocol for First Contact incorporates all of these principles. If a message is discovered, and confirmed as being from an extraterrestrial civilization, they have set out several steps that provide for proper notification. Transparency is at the heart of the protocol. The writers understand the danger in what might occur once a government is included in notification. Scientists, for the most part, have a commitment to openness. It is an integral part of the scientific process. Governments have a tendency to control information, often through secrecy and under the guise of military and public safety concerns.
Global involvement is an extension of the transparency idea. No one government should attempt to control First Contact. It needs to be a discussion held at the world level, and all nations should be able to participate.
After the big introduction we will need a well-considered plan. Alien First Contact will require a new type of diplomacy. In a SETI First Contact event we would probably have plenty of time to take up such matters. A signal could take years or even decades to get back and forth between Earth and the newly discovered civilization. Direct First Contact would require a much quicker reaction, and a much more conservative plan. Direct First Contact brings with it a set of threats and concerns. We would want to carefully control how further contact with Alien visitors might occur.
The idea of a carefully controlled process might seem to conflict with transparency. And indeed it could, if the reaction is to shut the world off from the diplomatic process. Both needs can be accomplished with a simple set of requests laid out nearly immediately. One body, preferably the United Nations, needs to control how further contact will continue, how information will be relayed, and what types of information we want to receive. We can be completely transparent if we set out our concerns right away, and provide a process by which diplomacy can begin.
Of course, aggressive First Contact may make these priorities unachievable. Still, we would be better off having as much openness between nations as possible, even in the worst case scenario.
Alien contact of any sort would be a milestone for human civilization. It needs to be an event, and a process, for all the people of the world.