Gowdas and Shettys are prominent and also dominant communities in the state of Karnataka. Where do these communities belong in the Chaturvarna System?
I have no idea about the Gowdas.
As regards the Shettys, they belong to the Vaishya Varna as per the Chaturvarna system.
The word Shetty is derived from from the Sanskrit word Shresthin (श्रेष्ठिन्), the same source gives rise to Seth (सेठ) of North india and Chettiar (செட்டியார்) of Tamil Nadu. They were traditionally merchants and engaged in the money lending/ banking business. Occasionally the term could also be used to mean the chief of a merchant guild.
There are many references to the money lending business in Kutilya’s Arthashastra as well as copper plate inscriptions of kings which I’m not aware about.