I mostly grow houseplant types and annual flowers, so I've never grown basil. But I agree that the leaves don't look healthy at all... the edges on most of them look like they're about to turn crinkly and pretty sure that's not just from insect damage. It also looks like it's not getting enough light, it seems pretty stretched out, unless that's just from the amount of leaves that have dropped.
If it's close to the window, is it too cold for basil? I know they don't like changes in temps much, nor do they like being wet. I've seen pots of them with multiple stems like that, but it definitely needs better draining soil and less watering. I'm not sure struggling with this one is worth it, you might save it, but it might be better to start with a new plant with better soil and conditions from the start.
And check other houseplants for signs of insects to get rid of them before they do any damage.
