In a pure free market economy, government would not take any steps to encourage the production of merit goods. This is because the government in a true free market economy does not intervene in the economy. However, there is no country in the world that actually has a pure free market economy, so I will assume that you are asking about countries like the United States whose economies are said to be free market economies even though their governments are significantly involved in making various economic decisions.
A merit good is a good that consumers in society do not value as highly as they “should.” In other words, it is important for them and for our society to have these goods, but people would not be willing and able to pay enough for them to make them universally available. An example of this would be education, which is very valuable, but whose value might not be obvious enough to individuals to induce them to pay its costs.
There are a variety of measures that governments can take to encourage the production of merit goods. The most direct method is to simply produce those goods themselves. In the US, the government provides public education to all K-12 students at no cost. A second measure is to subsidize the production of the merit good. The government could give tax breaks to those who produce the goods. The government could directly pay firms that provide the goods. All of these measures would increase the supply of the good.
The government can very indirectly encourage production of merit goods as well. It can do so by attempting to stimulate demand. For example, the government could give vouchers to low-income families who could use them to pay part of the cost of private schooling. The government could also impose price controls on the product, artificially lowering the price and, thereby, increasing the quantity demanded (though this might lead to shortages by lowering quantity supplied).
Thus, there are many policies that governments in a (mostly) free market economy can undertake to encourage the production of merit goods.