I don't pretend to understand why some web browsers (such as Safari) don't keep you logged in. There are many reasons, I'm sure, some of which are user-generated (e.g. you clear your browser).
I can tell you that I use the web browser Atomic Web ($.99, USD) and I remain logged in to site when I return to them.
I know, I know, it doesn't answer your question - but maybe it provides a relatively cheap workaround until someone else can explain why or how to make Safari do the job?
Sorry that I'm not of better help.
Marilyn