you would be looking at either
1. 4" or 6" recessed lighting
2. troffers (could be t8 or led) mounted recessed into the ceiling.
for example
http://www.cree.com/Lighting/Products/Indoor/Troffers/ZR-Series

The least expensive way with new luminaires might be T8 luminaires mostly around the periphery of the basement. Fill in between the joists where you need more powerful illumination. Program start ballasts would be best to maximize lamp life. Consider reflectors with surfaces at 90 degrees to keep most of the light off the walls and into the basement center, especially if the walls are dark.

If you want to do some on the cheap, people are always unloading old troffers. Check Craigslist, Freecycle and increasingly common building materials recycling centers. You could buy some program start ballasts on ebay and rebuild them with some new unshunted sockets. The latter can be a little hard to find.

If the joists and subfloor is dark, painting might help some, but would be somewhat of a pain.

FWIW, Philips has a new LED T8 drop in replacement tube which does not require any disabling of the ballast. Going price is http://www.treehugger.com/energy-efficiency/looks-fluorescent-tube-its-actually-41-more-efficient-led-lamp.html

http://www.buyriteledlighting.com/product/product&product_id=377

Here is a link that might be useful: InstaFit

Interesting product. It seems like it might be constructing a rather high maintenance light. If the FL ballast fails, you still have to replace the fluorescence ballast to get the light to work. It would be nice if they can make them so the FL ballast could be bypassed at that point. I suppose that is is possible that the power dissipated by the FL ballast is much lower with the LED assembly in there compared to with a long FL tube. That might extend ballast life because it will be cooler tending to minimize the impact. The ballast power consumption is lower with the LED retrofit than in typical T8, but would certainly be lower still with a system that cuts out the FL ballast entirely.

That makes me wonder if that stated ballast power in the is the fl ballast alone or both the fl ballast and the LED ballast combined. I also wonder what it does to the power factor of a good FL ballast.

Upside is that light quality looks pretty good. The price is good, a replacement ballast and lamp will run at least $13 compared to $26 for this device. Reasonably high CRI and good range of color are available. On the down side, the light output is weak compared to linear fluorescent, 1600 lumens vs, at least 2500 (normal ballast factor) for T8. That is probably lower output than ultra low ballast factor ballasts. I am getting old, so I would not settle for that kind of cut in light output.

T8 lifetime might approach the rated life of these (40,000 vs. 36,000 h) with 12 hour burns (programmed start), but the LEDs gain the advantage with frequent switching. (T8 drops to 30,000 hours at 3 hr burns), LEDs will be will fade more during rated life than the T8.