The reason we're ok eating steak rare, blue or outright raw is because the bacteria found on commercial beef sits solely on the surface, meaning that after a good sear in a hot pan (or chopped off in the case of tartare) the meat is safe to eat no matter how gloriously rare it is inside.
With minced beef, like the type you get for burgers, that surface is chopped up and distributed throughout the entire mass of meat. This means unlike steak, you do need actually need to to cook the meat to a temperature that renders the center safe to eat (unless you prepare the beef tartare style before mincing and forming into a burger). This temperature is reached at what we define as medium.
I love my steak as blue as I can get it, but with burgers medium is definitely the way to go. Besides being dodgy from a safety standpoint, the texture of borderline uncooked mince is not a pleasant one. Medium still has a nice rosey colour and plenty of moisture, the most important thing is searing as hard and quickly as you can to get a good caramelised crust alongside a juicy centre.
One thing beef does share regardless of form, is that cooking it well done is an abomination.