Maybe some physical damage to the plant stem or base from wind or animals? Birds will sometimes mess with plants to get at insects, or to stash food in them. Or rodents.

Could be damping off, sometimes that goes fast.

Could be a lot of things. If this is the only plant affected, just isolate it from the others or if bad enough pull it out and don't sweat it. Sometimes there's just that one plant that suffers from "failure to thrive" and you never really know why.

If there are a lot of plants in that one container it may be that they are competing for nutrients and water and that plant lost.

Basil in particular does not like a ton of water so maybe you've got a little bit of root rot going?

Never used a fungicide on herbs, so can't comment on that.

Basil likes a fair amount of water but it doesn't like to stand in water. They also aren't fans of their leaves getting wet.

I'd push your finger in to the potting mix in the area where your Basil isn't thriving and check for water - sometimes soil within different areas of a pot (particularly if it's a large pot) can become agrophobic or hold moisture better than other areas. After that, I'd dose the plant with Seasol (I LOVE this product and use it on all my plants as a tonic and as my first port of call if a plant that I've neglected...gasp! looks a little unwell) every fortnight and give the plant a liquid fertilizer.

Also I'd go through the plant and tip prune (pinching out the new leaves to encourage bush growth).

I don't use commercial fungicides even those marked organic and prefer to make my own - here's a link for homemade fungicides that I've used (I like the milk one best personally) http://www.abc.net.au/gardening/stories/s1484689.htm

Good luck!