Most of the time a failed power outlet or non working could be fixed with a very simple method. It depends on what kind of electrical boxes you have installed in your house. If you have the older electrical boxes system, it would probably be using the old round fuse plug system. In this case, you simply need to identify and locate the exact fuse plug which controls the power outlet along the side of your basement wall. Then you need to twist turn the fuse plug usually anti clockwise and remove the fuse plug. Install the new fuse plug by turning it clockwise. Be mindful of the power amp the old fuse plug uses, so you can go and buy the exact same fuse power amp for replacement.
If you do not know the exact fuse plug which controls the power outlet along your basement, you have to resort to the trial and error method. You will need another person's help to remove each fuse plug one by one until you find the exact fuse that controls the power outlet in your basement, while you observe to see which fuse plug controls which power outlet or location throughout your entire house.
For the newer electrical boxes, they don't use the old fuse plug system anymore. So you don't have to go out and buy those fuse plug. All you have to do is to identify the switch which controls the power outlet in your basement and then reset the switch by moving it either left or right and then returning it to its original switch position. Again with this method you will need the help of another person to check which switch controls which power throughout the entire house.