What part to use from fresh Basil leaves?
You can roll the leaves and thinly slice them (chiffonade), that's my preferred method. I've never had an instance where I cut basil and made it sour though.

I've never heard of boiling them, but I assume you could get the essential oils out with that method and could make an infused water. You can fry the basil leaves in oil as well to make a crunchy topping!