If you purchased EVE Online on Steam before May 30th, 2013, this behavior is correct and normal. Just log in with EVE Account name and password.
For EVE Accounts that were created through buying EVE Online via Steam after that date, the launcher should automatically log you into the EVE account that is tied to your STEAM account and should not display any entry fields for log in credentials until you click on the "Switch user account" link below the play button. If the launcher displays the log in entry fields, the STEAM authentication could not be completed.
In those cases, please ensure that your internet connection is up and running, that STEAM is not currently in Offline Mode and that the Launchers communication is not blocked by any security software on your computer.