Putting fans in a duct below the vents will help, but you'll need a fan at each vent. And it will also use additional electricity.
To keep the system passive (no electricity needed), you can create a "convection loop". You use the difference in weight of the heavier cold air upstairs, verses the lighter warm air in the basement to naturally move them where you want them. Gravity will be the driving force.
To better understand warm verses cold vents, it might be easier to think of them as two types of vents, "short" and "tall".
The "short" vents only go through the floor. They will let the warm air up.
The "tall" vents also go through the floor, but they have duct work that extends them almost to the basement floor. They let the cold air go down.
By having some air vents open to the warm air at the basement ceiling, and using other vents with ducts that extend the cold air vent opening down near the basement floor, you naturally prevent the cold air upstairs from damming up the warm air vents. That difference in duct length "steers" the air to form a convection loop. Then the upstairs heavier cold air will sink to the basement floor pushing the warm basement air up through the warm air vents.
If you have rooms that you close the door and you want them to stay warm, the closed door will short-circuit the loop. Therefore, you have to give each room their own convection loop. In other words, a cold air return vent and a warm air vent in each room.