Some would argue that the name of the currency is a proper noun, and it is (or should be) capitalized. But the name of the currency is rarely used. Let me explain:
If I have a dog named "Rover" and I hold him in my arms I can say "I have Rover". But if I hold a dollar in my hand, that specific dollar is not named "Australian Dollar", or "Dollar", so I can't say "I have Australian Dollar" or "I have Dollar". I must say "I have an Australian dollar" (or "I have one Australian dollar"). A dollar is what it is, not what it is named, and "Australian" is just an adjective. So in this context, lowercase dollar is correct.
There is only one currency of that name though. It could well be argued that it should be a proper noun (capitalized) when used to name the currency as a whole, rather than refer to individual units of it. However, even if we enforce that rule, we must still allow "I have 100 Australian dollars" instead of "I have 100 Australian Dollars", because what I have is 100 units of the currency, not 100 currencies all named "Australian Dollar". Also, it would be very odd if required to write "I have 100 Australian Dollars" but "I have 100 dollars". These issues are confusing, and blur the distinction between the supposed proper noun name of the currency and the common noun term(s) used to refer to amounts of the currency.
The name of a currency is used less often than referring to amounts of that currency. But see, for example: "The Australian Dollar (AUD) is the currency unit used in the Commonwealth of Australia...". There it is treated as a proper noun, but not everyone does this; for example: "The Australian dollar, denoted by AUD or A$, is the official currency of the Commonwealth of Australia". Perhaps the lowercase usage is sloppy, but it is not wrong if one interprets "The Australian dollar" not as a proper noun, but as meaning "that dollar which is Australian".
Also realize that terms that are very popularly used can lose their capital letters, even if they should have them, due to careless usage and forgotten origins. See for example on Wikipedia: List of generic and genericized trademarks.
Ultimately, trying to capitalize currencies can cause more confusion than it solves.
For all these reasons, people aren't too fussy about it.