Etsy doesn't have an affiliate program. Why do they still have an API?

What's the point of Etsy providing an API when they don't have an affiliate program. No site will be able to monetize traffic sent to them.

Or do they do deals on an individual basis with developers who use their API for mashups?

The Etsy API is primarily for sellers (instead of buyers/clickthrough). As an API developer, you can make tools to help post and manage new products. There's also the ability to run analytics and even create aggregate shopping experiences so that you can host your shop on your own network, but let users browse any product.

Buyer tools are also in the mix but have very little value, even seller tools will rarely be needed as it's less likely for businesses to have a huge inventory of hand-crafted items. That said, it is a bit of an issue. Sure you can create apps to monitor your sales or browse Etsy niches,... and you may sell the app itself... but it would make much more sense to have the ability to put a small markup on the products sold directly by your app or split some of the transaction fee profits with Etsy.