Your links:

I don't know what's wrong but a guess would be it's a combo of not enough light and too much water.

I'm not sure what it is either, but what kind of lights are you gorwing them under? You might want to try some T12 or T8 flourescent lights. Also, I have some other plants doing pretty well under a 75wt compact pigtail flourescent, which is equal to 300 watts. I only grow basil outside though because it thirves outdoors where I live. I'm not an advanced garder or anything, but all I can recommend is when gowing indoors try to mimic the plants natural outdoor habitat it thrives in. I'm in a southern Texas costal area. It's very HOT (about 100F every day), very HUMID (about 60% day 90%+ at night), and I water them a lot but mainly because of the heat. Also have you put any fetilizer in them? I know some people don't with herbs, but I would give them at least 1 shot since they're looking sickly.
Lastly, on the cutting you're trying to root... I always root my cuttings in peat pellets. I see a lot of people on these boards hate them, but I love them and have always had great success with them. They were recommended to me by an extremly tallented gardener. Make sure you got a stem and not just a leaf. There is rooting hormone gel or powder. I take my cuttings, dip them in root hormone, and then stick them in a peat pellet. I set them in trays with a clear lid over them and place under T8 flourescent lights until the roots start coming out. I've made 3 basil plants off my 1 original 1 that are all flourishing.
I have no idea what your climate is like right now, but you should go to a local nursery or growing supply store and even take your plant or pictures. You could also check the pH of your soil and water to make sure they're okay. Where I live out water is alkaline, so I put pine needles in my pots to balance it out. Good luck with your project.

thanks Lmoon310 for the advice~

as for the lights....I just use my ordinary light from my room(I have no idea what bulb its called)
our climate here's a little more or less than 30�C(around 86�F?, humidity: day: 70%+, night: 80%+)

I haven't given them any fertilizers, yet...and I just used plain tap water
for the cutting, I just cut a stem(with some leaves) and just plainly placed them on a glass of water, I didn't use any rooting hormone

they've been outside every morning since 3 days ago, and I put them indoors at night since our weather at this time of the month is very tricky
I'll water them next time w/ liquid fertilizer to see if it can cure them~
again thank you very much!

so, I'm guessing that those brown spots and brown stem are still cureable?