start minimized browser instance with Karma

Building off of @Jeff Sheets great answer, you can position the window off the screen using the --window-position flag, so it's at least out of your way if not actually minimized. It's also good for multiple monitor situations:

    browsers: ['Chrome_small'],

    customLaunchers: {
      Chrome_small: {
        base: 'Chrome',
        flags: [
            '--window-size=400,400',
            '--window-position=-400,0'
        ]
      }
    }

The best I've found for Chrome is to set the window-size. It still maximizes but at least you can quickly hit the Restore button to resize it to a small window. I couldn't find an option to not open maximized http://peter.sh/experiments/chromium-command-line-switches/

Here's the Karma config:

browsers: ['Chrome_small'],

customLaunchers: {
  Chrome_small: {
    base: 'Chrome',
    flags: ['--window-size=400,400']
  }
}

Firefox

No, you cannot at the moment. There's an issue on the repo for that but it hasn't been closed yet.

While it is possible to pass some user preferences, there's not yet the option to pass the -tray command line option described here.

Chrome

Update 2022; below causes errors, like:

Chrome failed 2 times (cannot start). Giving up.

You can definitely add flags to the Chrome launcher configuration.
I'm not sure you can do what you are looking for tough: the only thing I found is the --no-startup-window flag.
Add it to the launcher configuration section and let us know if it works for your purpose.
Have a look at the launcher documentation to see how to add flags.