Just wanted to say, that I changed my password thinking that I would see the, "remember my password" box. I didn't. Grrrr. I think it's nonsense that by default it runs steam in the background. I hope I can get around this.
If Steam is running without asking you for a password at start, and going straight to the background, you're good to go anyway; remembering your password is the default behavior. If you don't like that, there's a box in Settings that says 'Don't remember account information on this computer.'