It does depend on what exactly you mean by "old". The final two series of 10/- notes, isued 1953-1966 with Flinders on one side and Parliament house on the other, are common enough and if worn or dirty they should probably be sent to the bank since no-one would want to pay more than a dollar for it.
Older notes, from the time of WWII and King George VI (which are more likely to be present in America, since so many American soldiers served over here during WWII) are more valuable and should never be banked unless they're literally coming apart, as you should be able to get at least a few dollars for them.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis