Will adding a cool air return in the basement help with summer AC power usage?

Furnace installers and/or home owners will put a registers in the duct work to provide some heat or A/C to that area. The number of these supply and return registers is determined by the typical use and desired temperature of this area. I recommend that all basements should have at least 2 supply registers and 1 return register in the basement/equipment area just to keep that area slightly warm or cool and the air fresh. If you are going to finish that area to be used as living space, then more supply and returns will be needed. Also you may want to close the supply registers in the summer when using the A/C since cold air is heavy and will move to the lowest level in the house. Leave the returns (mounted at the floor level) open to circulate that air all the time.

We must assume that whoever taped the supply vent shut did so for a reason--probably it was too cold in the summer (and maybe or maybe not too hot in winter). Too much conditioned air was going into the basement space and not enough into the upper floors.

If you put a return into the basement and open the taped up vent, then even more conditioned air will go into the basement--the opposite of what you want. First try leaving the basement as is and opening up dampers or restrictor vanes in the supply vents to the upper floors to allow more flow into the upper floors.

Does your system have a return vent system? If so, if you later want to provide airflow to the basement you could install a return vent and connect it to the return system. But if this would be difficult it might be sufficient to simply have a return air connection from the basement to the floor above. Of course, this might reduce audio privacy depending on how it was done.

I have installed air intakes in basements many times. The natural cooling will reduce your AC bill. If your stairwell to the basement is open you don't even need registers down there and the furnace will push the cool air through the duct work. By installing dampers you can change the flow from different levels by season. My last home I did exactly that and it greatly reduced my cooling bill and I had a wood stove down stairs that shifting the dampers for winter saved me a huge amount on heating and all I did was change the volume of air removed from upstairs or downstairs. Upstairs open fully in the winter downstairs in the summer. It worked great and saved quite a bit while keeping both zones more comfortable.

Get one of the AC units that have wheels and sits on the floor and exhausts out the window and use that at on the top level. Your Central AC wont work as hard. I have just done this and my electric costs have actually gone down.