Libertarians would say no to this question solely based on the premise of using public funds to fund anything at all. Supporting families and encouraging childbearing are worthy goals. Yet just as libertarian Christians reject caring for the poor or sick with stolen taxpayer money---or with borrowed or inflated currency---they also reject funding pro-family programs by such means.
Beyond this ethical concern, pro-natalist subsidies create moral and economic hazards. When government redistributes wealth, it distorts the natural market signals families and communities rely on to make sustainable decisions. Subsidies may encourage people to become parents before they are ready, or even incentivize single parenthood, trapping households in cycles of poverty. Financial incentives may also attract individuals lacking the maturity or resources to raise children well, undermining rather than strengthening family life. Free money comes without accountability: it does not ensure that resources are used responsibly or in ways that genuinely nurture children.
By contrast, private charity and church support are embedded in real community. These contexts provide not only material help but also mentorship, accountability, and relational structures that build families up in ways government programs cannot replicate. Christians are called to bear one another's burdens---but doing so through coercive taxation undermines both liberty and the integrity of family life.
This question is from Faith Seeking Freedom: Updated & Expanded, available in paperback, PDF, and Kindle.