Best lighting option for basement
Recessed fixtures would seem like a perfectly good option, since they won't project down below the ceiling. If you wanted to get complicated (a bit of drywall work) you could actually bump them up into the ceiling a bit.

LED (generically) or "relatively efficient LED" more specifically without getting any more specific (it's a moving target anyway) is an excellent idea - you can light the whole basement on what one or two old-style bulbs took for power, and bulb replacement is a non-issue. Just do keep an eye on the lumens-per-watt figure - some are just selling not very efficient fixtures that happen to have LEDs in them in the hope that you'll see LED and ignore lumens per watt.

Given that you have the open side to fish to, I'd just wire them up and call it a day. If you also replace the switched fixture they'll connect to, you might consider dimmable LEDs (many are these days) and an LED dimmer. OTOH, if you can't think of a reason you'd want it dim/half-bright in the basement, leave the switch.