So does Chrome.
And Firefox.
Moderator Note
Just to let everyone know, we are aware of this issue. It is affecting us as well.
There’s probably nothing you can do on your end to keep yourself logged in. You will just have to keep logging in to use the board.
Sorry for the inconvenience. The tech guys are aware of the problem. Hopefully it will be fixed soon.
The session id’s (or whatever they are) time out. Mine only lasted a few hours, and then I was logged out again.
It doesn’t work at all on Silk on a Kindle Fire. I enter my username and password, it says I’ve logged in, then it gives me the same default homepage with the login prompt. I’m posting this from my iPhone.
I’m thinking of changing my screen name to Dido, so often have I said “remember me” in the last 24 hours.
Works for me apparently.
Single quote works, but there’s a problem with the multi quoting. You have to copy and paste additional quotes.
It would sure be a lot easier to type in on every single visit than “Biffy the Elephant Shrew.”
Some web browsers can auto-fill login information. Chrome remembers what I’ve typed the user name field for SDMB, so when I click there, I can a drop down of past values. (They’re all my name, with different capitalization).
I still click the Remember Me box, just in case… 
Another one having trouble staying logged in. I also can’t get into the Thread Game threads. I can see 'em but I can’t post in them.
I am now able to access my User Control Panel without logging in again.
Hopefully soon. I’ve given up on clicking the Remember Me box, and I’m starting to forget who I am.
It’s your speeding ticket. We’ve decided to slow down your posting rate, Professor.
To, say, under 30 posts per day for you.
It is to laugh. 
Actually, I find I can get into some threads (Cryptic, Botticelli), but not others (Baker’s Dozen, Movie Titles). Which is unhappily curious.
Quite bizarre.
Safari does have autofill, but for some reason it doesn’t seem to apply to the SDMB login. (I also have Chrome, but I never use it.)
Safari on my iPad works great for me logging in here.
No, you have to keep on clicking it, because some day, without notice, the feature might be re-enabled.I I refuse to upgrade from an older outdated version of FireFox, and it autofills. I just need to enter “j” and it fills in username, then password fills in automatically. So just one key entry and two mouse clicks. I guess I can live with that if that is the price of freedom from digital terrorism.
TLTE:
The downside is I have to keep backing up through SDMB, and can’t just re-hit my bookmarks button to restart the page.