bluemitten, unless your blog is doing something very unusual I don't see the problem. Yes, clicking on 'Login' will take you to the login page but you still need a username and password to actually login. If you have a good password, I don't see the problem. Are you saying that you click 'login' and automatically get admin access? If so, does this happen on computers other than yours? Or maybe you are concerned that your username and password are filled out for you? That is the browser, not WordPress. Browsers can, but can be told not to, save data like that but it is only a problem if other people have access to your computer. Again, does this happen on computers other than yours? I don't understand this line-- 'my blog is completely open to whoever clicks login'-- or how you came to that conclusion.
However, if you don't like the Meta widget-- I don't, actually-- just remove it. Go to Appearance->Widgets and drag it out of the sidebar.