On some smaller resolutions the Starbound launcher is to large to hit the play button as it extends past the edges of the monitor. To play I have to change resolutions and drag around to find the button and then play it and switch back. Please add better resolution support or atleast a window resize option.
my friend is having a similar problem, they are using a mobile monitor and they can't play. please at least allow us to resize the window.
Meanwhile you can launch the game without the launcher. http://playstarbound.com/support/#launchdirectly