I googled and it seems that basil does turn woody after some time but the leaves on these browned stalks look really unhealthy, like I haven't been watering them. Also, the stalks got thinner after they've woodened.
I sprayed them with garlic because I thought it might be a fungus but it doesn't seem to help. What's going on?
Picture of stems at the base:
Picture of individual stems with wilted leaves:
It's damping off disease - hopefully, you're not overwatering now, and the rest of the plant still looks healthy, but you may find, as its a fungal infection in the soil, that the rest of it goes the same way, though that's uncommon in more mature plants like yours. Good airflow round the plant helps to keep this at bay, but there is no effective fungicidal treatment once its got a hold. It usually happens when plants are at the seedling stage, and is associated with using unsterilized soil or soil in dirty pots. The link below is mostly about growing sage, but if you scroll down to the bottom, you will see an image that is almost exactly the same as the one you've posted of the three affected stems
http://www.onthegreenfarms.com/fruit-vegetable/how-to-grow-organic-sage/
More info regarding damping off disease generally here
http://www.planetnatural.com/pest-problem-solver/plant-disease/damping-off/
