Basil seedlings gone weak and falling over.

[veggiechicken]

If one lot is OK and the other not - have you overwatered the floppy ones?

[Two_Sheds]

You may think it's bright "enough" but it might not be. Look up the lux needed for growing plants (esp. Mediterranean plants like basil)

"On a summer's day ... we may have up to 16 hours of daylight at 100,000 lux. In winter an 8 hour dull day will give less than 5,000 lux and indoor lighting rarely exceeds 500 lux."

Your seedlings may well have S.A.D.

[Jeanied]

I agree with all the above and would add that basil is particularly prone to damping off - so sow thinly in new compost, at the right time of year, and make sure the trays and watering can / bottle are scrupulously clean - and only use tap water.