The simple answer is that they did not. Ancient governments simply did not have the huge bureaucratic capacity that modern states did, not even the capacity of eighteenth century states. So by and large they did not know how much money was in circulation, and while there was some understanding that the amount of money mattered it was not particularly well understood (some critics of macro-economics would claim it is still not well understood).

Ancient economists still argue today about whether ancient people thought in these terms. In the early twentieth century historians broadly just superimposed their own contemporary understandings onto ancient governments and their economic policy. There was considerable revisionist blow-back against this from various fields in the middle of the twentieth century, key thinkers are Karl Polanyi, Marshall Sahlins, and Moses Finley. If you want to know my thoughts on the last of these, Moses Finley, who falls closest to my own expertise of ancient numismatics you can see a summary I gave at https://youtu.be/TscLNCYj1-w . There have been reactionary trends against this trying to argue that ancient people did think more economically than people like Finley give credit for and I have read and heard good arguments for that, but this is not a case of the truth falling somewhere in between, economic thinking seems to be the exception rather than the rule.

With regard to inflation as a concept in the ancient world it is very complicated. We simply do not have very much information about prices, money supplies, etc. Even for coinage, the type of money it is easiest to measure, it is worth remembering that a great deal of basic data is lacking. Outside of the United Kingdom there are virtually no effective schemes for recording single finds of coins (look to the work of Roger Bland and the Portable Antiquities Scheme if you are interested). For most of South and Central Asia we still lack basic type catalogues for most coin issues, and even for better studied areas most coinages lack die studies (a numismatic technique for reconstructing the volume and rhythm of minting). So in the absence of information about production or deposition estimating circulating coinages is exceptionally difficult. If you are interested in somebody looking at this problem there is an interesting book by Duncan-Jones called Money and Government in the Roman Empire. I would not endorse every aspect but it will give you a sense of how thin the evidence is in even the best understood periods.

And finally I would caution against thinking too much in terms of 'gold'. Most ancient coinage was silver and just as not all coins were money, so not all money was coin. The ancient world had a very wide variety of forms of exchange, and only a very small proportion depended on coined money. So it is not at all clear how making more money would actually effect things. A final recommendation of a controversial book, The Big Problem of Small Change. It is not a good example of applying economic theory to historical data, but it does have some quite detailed discussions of coined money in the medieval world - and the notable thing is that people spend far more time complaining about governments not making enough money than about their being too much in circulation.