That’s a good question. Currently, we can know that by reasoning about the logic: every miner validates every transaction, and each transaction comes with a zero-knowledge proof that it doesn’t violate conservation-of-money (i.e. a proof that the money coming out of the transaction is ≤ the money going into the transaction).
This reasoning depends on the soundness of the zero-knowledge proofs. If you could somehow get the miners to accept a transaction that created new money — if you could somehow forge a zero-knowledge proof or defeat the zero-knowledge-proof-verifier software in the miners — then you could counterfeit money.
In the future, I’d like to add another layer of detection of this: I’d like to have a regular (e.g. monthly) “accounting period” in which the existence of money must be publicly revealed in order for the money to remain valid. This would not damage the user’s privacy—the amount thus revealed would not be linked with the owner’s address or with any other transactions, and even the amount itself could be obfuscated by splitting it into 2 or 3 random sub-amounts before revealing it. But, it would allow everyone publicly to count the sum of how much money was revealed in that accounting period, which would give us a backward-looking detection of whether counterfeiting had occurred during that period.
To change the protocol to require this would be a “forking” upgrade, in the sense that it would require all users who want to use the new protocol to upgrade to new software, and if people didn’t upgrade — either because they weren’t paying attention or because they preferred the original protocol — then they would spawn off a parallel chain-fork.
A protocol upgrade like this could also institute expiration of old spending keys and old unspent notes, meaning that your wallet has to regularly freshen your Zcash balances in order to keep them alive. For example, the protocol might say that money which has been lying idle for more than a year expires and is not longer accepted by nodes. Wallets would refresh money once a month, so that people wouldn’t lose their savings unless their wallet got disconnected from the blockchain for more than a year. If you wanted to store that kind of Zcash in a long-term, off-line vault you’d have to bring the wallet out at least once a year to let it connect to the blockchain and refresh the money.
One advantage of this “old-money-expiration” feature would be that the current state maintained in the blockchain could be garbage-collected, greatly reducing the scalability problem and allowing substantially faster performance. Another would be that everyone would be able to tell — in addition to the upper bound on the monetary base mentioned above, which detects counterfeiting, a tighter upper bound on the monetary base, which detects lost spending keys.