Go into FactionDefs.XML in Galactic Civilizations III/data/Game in wherever Steam installed the game on your computer (by default, C:/Program Files (x86)/Steam/SteamApps/common/). Then look for the lines that have the tag <StartingShips>(some ship type here)</StartingShips>.
It looks like these are defined separately for each faction rather than being something common to all, so you'll need to go through each faction entry. Fortunately (or not), they're all in the same file. I think FACTION_DEFAULT is probably the template for custom races, by the way, so removing the <StartingShips> lines from that I think will make it so that custom factions won't have starting ships.
Also, I'd suggest copying the file, modifying the copy, and placing the copy in /Documents/My Games/GalCiv3/Mods/NoStartingShips/Game and enabling mods in game rather than editing the base file, and I'd also suggest creating a backup file whatever you decide to do. You'll need to create the NoStartingShips directory (feel free, however, to name it something else).