To close or not to close the basement door

I apologize if this is the wrong subreddit. I figured this is the best place to ask. I'll make it short and sweet...

We have a tri-level home. You enter from the front door to the main level, then there is a half flight of stairs up, and a half flight down into a daylight basement. We have central air, but not central AC. Since the basement stays much cooler than the rest of the house, we sleep down stairs during the hot days. Our return vent is on the main level, which is much warmer than the basement. Since theoretically the the return vent is piping the warmer air from the main floor down into the basement (we keep the fan running to cycle the air), should we keep the door leading down to the basement closed, or open for maximum cooling in the basement?

Please let me know if any of that is confusing or if anyone has any questions. Thanks so much!

submitted by /u/teejaydub
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