I will give you the same answer I gave when the same question was asked about Starcraft 2:
Once upon a time, there was a way to make World of Warcraft (Blizzard's other giant game) remember your password. You could edit certain files to have your password be filled in automatically.
Blizzard specifically disabled this ability, by removing all ability to change the login screens. They did this in part to prevent spyware/viruses from editing them (to steal your password), and in part because many people play Blizzard games on shared computers and in cyber-cafes, where saved passwords could lead to all sorts of mischief.
So, you can't directly save your password for a Blizzard game. At least not until 2027 when you can install ActivisFaceBlizzTwitterBook directly into your cerebral cortex.
Carl's answer mentions some ways to get around this; however, save your password into a macro hotkey program at your own risk.
Update for D3: Although your password must be entered each time, Blizzard has a new "intelligent login" feature related to entering your authenticator code. If this feature is enabled in your Battle.net account (enabled by default), then when you consistently log in from the same location, you should not have to enter your authenticator code on every login; you just have to enter it once a week and/or if your IP address changes. This means you can at least enter the password from muscle memory and not have to access your authenticator (after finding the darn thing on your desk of course).