When Jubal Early confronts Simon Tam in the corridor, he tells Simon, "The warrant doesn't specify any particular need for you to be alive." In the opening scene, when Jubal looks at the warrant, it clearly says, "Wanted Alive" under River's name, not Simon's. The Alliance considers River a valuable asset, but Simon is just a fugitive, and probably wanted dead or alive. So what Early says is correct.
(at around 12 mins) While Kaylee is revealing River's extraordinary killing skills in the common area, you can see a crew member on the left side of the screen, bent over the counter reading a binder.