Trying to strike cutting in water is not a method recommended by horticulturists. It's far better to put your cutting straight into soil. Point is, water is not a natural growing medium for plants like basil, and the risk of the cut end rotting is very high. Your cutting is unlikely to produce roots now that cut end has turned brown, though there is still a slim chance.

Try cutting off the rotted end, and planting the cutting into some potting mix - or straight into your garden. It's more likely to produce roots, and those roots can then access nutrients from the soil. Remember, water has no nutrients.

Remember, 2 weeks isn't a long time in terms of the growth of babies!

I've never had much luck rooting basil in water. The few times I got roots they were pretty spindly and weak.
Basil roots well in a half and half mix of perlite and vermiculite that is kept damp, not wet. If rainwater is not available for watering the mix, I use tap water that has set out for 24 hours (dissipates some of the additives). The roots are stronger, too. I don't add any fertilizer.
Would your neighbor give you fresh cuttings?

It can be done. I have done it myself.
I like Thai basils but did not want spend $4.00 for a small plant and could not find seeds. So I just bought a bunch of Thai basils, picked the real fresh ones, trimmed most of the leaves,... rooted them in water. As soon as I saw tiny roots I planted them in pots and eventually in the garden. Now I have a lot of them.
Some of the cuttings may rot, some wont.
Put a piece of aspirin tablet (ascorbic acid) in the water to help prevent rotting.

I put broken branches of basil pushed into nearby soil where basil was growing. The area was well watered a few weeks and at first the broken limbs wilted a bit, but soon recovered and became new plants.

Mango, are seeds a possibility? Basil seeds sprout easily.

I've rooted basil successfully many times in water. If you don't see some roots forming in a couple of days, cut off a bit of the bottom and try again with fresh water. If they're going to root, they will do it quickly.

It is helpfull to cut the stem end with a sharp knife or razor under running water and rightaway put the cutting where it should be. This will allow the cutting to drink better. If you want to get very technical about it, you can use pre-boiled cooled water to minimize the possibility of rotting. Furthermore, add a quarter of Aspirin tablet to make the environment slightly acidic.
Most bacteria cannot be active in acidic medium.
The cutting has enough stored food/energy in it to start a root. So it is immaterid if rooting is done in water or soil. What is important is "after root growth" that the cutting can benefit from nutrients from the medium.
That is why , it would be better to pot the cutting as soon as you see roots.
cyrus

Ive rooted a lot of cuttings of Holy Basil ,genoveve basil & cinnimon basil in water

Take a fresh bunch of basil, trim off the bottom leafs so there is a long stalk.
put in a cup with just enough water to cover the bottom portion of the stalk ( an inch or two)
place a plastic bag over the basil, rubber band the bottom ( I took the rubber band off after a week to get the air flowing)
place in in-direct sunlight
wait two weeks, changing water as needed
when I used this method I had a cluster of white roots at the bottom, and roots protruding out up half the stalk!
just planted them today, will see if the roots are strong enough for soil!

cyrus-gardener - I don't think aspirin is Ascorbic Acid. That's Vitamin C. Aspirin is Salicylic Acid. I would also recommend rooting in a potting mix or soil, not water. Roots which develop in water are not the same as proper roots and frequently break off when you eventually have to pot the plant. Save a step and just start in soil. You'll have better luck cutting just below a node (where a pair of leaves comes off).