How do I get the cold air from my basement upstairs?

If your furnace is in the basement turn the blower on, the fan it will circulate the cool air upstairs

Put a box fan at the bottom of the steps and send that cool air upstairs!

I always use a fan blowing up the stairwell. You could also turn the pilot light off on the furnace and just use the blower

If you have a forced air furnace, keep the fan running on your furnace 24/7 regardless if the central air is on or not. you can pay a HVAC installer to install dampers in the air ducts, so you can reduce the flow of force air feeding the basement by adjusting the dampers. another fix is have them install a second thermostat to the HVAC system and split the ducts into 2 zones. Most control boards in the furnace controllers do have the electronics for multiple zones, it is they just are not used in the average home. Another option is to close the vents downstairs during the summer, but I do suggest you do not close them all the way. I did that once year and because there was no longer tons of cold air blasting out of the ducts, the downstairs developed a dusty mildew growing on nearly piece of furniture, so do not completely block them allow some air circulation so the AC system can pull out the moisture in the air. Another simple fix is to keep the basement door open and place a couple of fans at the base of the stairs and at the top to push the cold air upstairs.

Does your thermostat have a setting that says "Fan"? Most thermostats have three settings - Heat, Cool, Fan. However, if yours doesn't have such a setting, I would do what Lori and Jean are suggesting and that is, use a regular floor-standing fan to, hopefully, move some of that cooler air upstairs.