Leopard and Snow Leopard both ask for a password to open a downloaded application for the first time, and if they don't know it they can download what they like, but it'll never run until the password is entered.
This isn't true. I download applications all the time and all Leopard says is 'This is a downloaded application from the internet. Are you sure you want to open this? <Cancel> <Open>'
You may have your security settings set lower than mine, but I'm naturally over-protective of my computers when it comes to the Internet. If I don't enter my admin password it won't let it run first time.