I have a 1st floor kitchen sink directly above a basement sink. When the 1st floor sink is full, it drains quickly and backs up in the sink below - which then drains relatively quickly. No other problems here - the dishwasher drains fine, general use is fine - it's only when there's a full sink of water going down that it backs up.

I have a 1st floor kitchen sink directly above a basement sink. When the 1st floor sink is full, it drains quickly and backs up in the sink below - which then drains relatively quickly. No other problems here - the dishwasher drains fine, general use is fine - it's only when there's a full sink of water going down that it backs up.

You're partially clogged in the drain line downstream from the basement sink.
To clear this line you must snake the from the kitchen roof vent. Put our enough cable to get down to the bend at the base and about 20 to 25 feet more to get from the branch to the main. While it drains away now this will only get worse. Better to take care of it now. Good luck, Tom

It's a partial clog at some point past where the basement sink goes into the line the kitchen sink drains into.

You'll want to snake it from where the basement sink drain goes into the wall or floor.

Lol, speedball1, you hit submit a moment before me :p
(I'm not a "repeat the answer for points guy")
Happy snaking, Gandalf!