The Starhawk multiplayer beta has officially been made open to all PlayStation Network users as of yesterday. While it had previously been available to some people who purchased Uncharted 3, and to a number of randomly selected people, the beta is now available to everyone else. There isn't anything fancy you have to do; just head to the online store and start playing.
Harvard Bonin, the senior producer of Santa Monica Studio, also had this to say regarding the beta:
Starhawk is a third-person shooter that is currently being developed by LightBox Interactive and Sony's Santa Monica Studio. It is set to be released exclusively for the PlayStation 3 video game console, and it will be published by Sony Computer Entertainment. The game looks to be a spiritual successor to Warhawk, a similar title which was released almost five years ago. This new game will support up to 32 players online, which certainly seems impressive.