I don't know about Australian banks specifically, but it's fairly hard to get a USD debit card attached to a bank account anywhere outside the US. Even in Canada the best the banks offer is a USD account and then a USD credit card that can be paid from the account. Offshore banks (IOM, Channel Islands etc) were forced to withdraw USD debit cards after the US "patriot" act came into force in 2001.
The work-around is to receive USD payments and make USD payments from Paypal, as far as possible. Never keep a large float there (Paypal are unreliable, to put it mildly), but it's an option. Maintain the balance in USD, don't convert it.
"Travel money" debit cards issued by Australian banks in USD in theory should help with the payment bit, but really only for something that doesn't have a verifiable address attached, which rules out a lot of online use. Might be worth checking with your Ausralian bank to see if you can receive money into a USD account there by wire and then issue a travel money card from the account without going through AUD. I doubt any are that flexible, but worth a try.