How do you audit that the issuer is printing coins backed 1-to-1 by USD reserves?
You’re asking the wrong question, since DAI isn’t backed 1:1 by USD reserves like fiat-backed stablecoins (e.g., USDC). It’s a decentralized crypto-collateralized stablecoin—it’s backed by a variety of assets, including cryptocurrencies and real-world assets, with mechanisms to ensure over-collateralization.
I’m on mobile and it’d take forever to explain the entire protocol to you here, so I recommend you read the white paper [0] if you’re interested in understanding how it all works.