Yep, this is the major downside of no user accounts. For this to work it will require signed messages. We'll try to make it as easy as possible, but would be signing something like "doge=xxxxxxx, ltc=yyyyyyy" from your BTC account, so we can verify it was you. I expect a huge number of "HOW DOES I SINE?" emails, and a lot of "My address is at my exchange, so I can't sign" problems, so we'll see how that goes...
Maybe the answer is to add optional user accounts? If you want to use the pool as it is today with default payout options, you would not need an account. But if you want non-default payouts, you'd need to get an account and set up your profile and payout options.
I know that adding accounts has its own set of complications, like database schema changes, email verification, SSL certificates (and avoiding Heartbleed), two factor options, etc. But it's probably simpler to support because users are familiar with accounts and managing profiles and options. Also, you can build new features on an account system later, like hash rate drop alerts, etc.