You can circulate the air if you want to balance it out, but the basement is going to get colder again.
If you're talking about saving energy:
At about 3pm circulate the air. That's a little before your AC is going to start experiencing it's highest workload.
At around 6pm or when ever, stop.
Try it for a couple of days with just a fan. If it's a noticeable difference and you like it, you can get a vent installed that pushes up from the basement, and another somewhere else that just goes straight to the basement. You can put the fan/blower on a timer. I'd recommend one of those "smart plug" things, they work as a timer and you can also controll locally from your phone.
But if you're circulating air 24/7, it's just making your AC cool even more air.
So you just want to use it to dump a bunch of cold air when you need it most, and then let it naturally cool down the rest of the nigh/day.
Whether or not this adds up to more than negligible benefits for energy use...
I have zero idea.
But it's essentially just an inefficient heat pump. The theory behind it is sound.