No, you don't get a menu where you can select everything to your liking, but with the officer requisition store next to sickbay in Earth spacedock and the amount of green and better officers you get as rewards just from leveling up, you're bound to find a race/gender mix you like. It's not perfect, but I actually like leaving some aspects of my crew to chance. I might never have chosen a Saurian if I'd been able to tailor every aspect of my crew myself. Now I'm glad I gave her a shot and I love my new chief engineer.
Considering you can change everything else about them, you might find you enjoy rolling the dice and seeing what Starfleet hands you.