@sinder
Falling over and drooping like you listed off will not be caused by lighting. That could be caused by several things - I'd look to over-watering and fungus as concerns.
Can you easily get your hands on basil seeds in the types you want to grow? I don't know where you are but I am guessing northern Europe (and your English is better than most native english speakers when they're on forums, good job)
Maybe the style your trying to grow the basil doesn't work for you. Try a few changes in addition to trying to keep this one.
I view basil as a fast turn short duration plant. It's easier for me to grow 20 or so plants for 5 to 8 weeks then put the ax to all of them and replace with another newly started 20 than it is for me to grow one big bushy plant. If you start successions of plants every week you'll have plenty especially if you're growing. I'm starting another 30 basil plants today and transplated 30 others out about a week and a half ago - I'm about 8 or 9 weeks from my first frost date.
1 - try a wide shallow bucket of soil and sow a bunch of seeds and see if that works for your. I'm thinking something like a breadpan. On the order of 250 mm x 120 mm and 80 to 100 mm deep. Start 10 plants in there. This is far tighter than anyone outside would ever grow but it is a test and you'll harvest them in 5 weeks when they're small anyway.
2 - look into Kratky hydroponics. Go to your favorite search engine and copy paste in "Kratky Hydroponics". This style is great for short lived small plants like lettuce and basil. It will cost you a small amount for nutrient, a small plastic storage bin, and some type of net cup.
Frankly I grow lettuce and basil using this fancy sounding Kratky method in old glass jars on my kitchen window sill. A pasta sauce jar is perfect for my little net cups and I drop a plant in there. Simply top off the nutrient liquid weekly.
read over this article:
Kratky hydroponics
read over this article :
Growing Hydroponic Basil