Personally --- I usually route power from the service panel to the basement lights since the panel is usually located in the basement and reduces the draw on the circuit upstairs.
If you decide to go this route and is the way mentioned above --- you will need to disconnect the cable that feeds the lights from an upstairs circuit. Find the source receptacle and shut the power off.
Disconnect the conductors from both the source receptacle and the light fixture ---- twist the white and black together cap with a wire nut --- wrap electrical tape around them --- done this way at both ends of the cable.
Or --- cut the conductors that feed the basement lights flush inside the upstairs receptacle box --- cut the whole cable in the basement flush with the flooring above.
Since you have receptacles already in the basement it would be simple and easy to cable a supply from a receptacle to the switch --- a 14/2 from the switch to light 1 -- 14/2 from light 1 to light 2
Disconnect the cable from 2-3
The same applies to the other two lights.
Supply to the switch --- 14/2 from the switch to light 4 --- 14/2 between 3 & 4
I like to have the light circuit seperate from receptacles . This way if a device plugged into a recetacle trips the breaker it doesn't affect the lights ---- you're not roaming in the dark trying to get to the service panel.
This way will require a breaker to added to the service panel and a cable run.
It could be wired as Jack's diagram shows --- just make sure the first box is large enough to accomodate eight conductors.
The choice is yours.