Once you're logged in, it generally sticks unless your cookies get cleared or some other wildly technical thing happens. My logins last for a couple of weeks minimum. Problems are often because 1. you've set your browser to clear cookies when you close the program or 2. a third party add-on is causing trouble.

Are you using a login/username switcher add-on? If so, disable it, log in through the dreamwidth page (https://www.dreamwidth.org/login.bml) and make sure you've ticked the "remember me" box. See if your logouts keep happening without the add-on.