The problem for the Reagan people is a simple one. How do you transfer a mass of responsibilities from the government to the private sector?

The tools also are simple ones. They include a stick and, sometimes, a carrot. With these tools, they can hit a program here and shake a tax incentive there, and maybe, just maybe, it will all work out.

This is the transfer-and-pray theory in action in Washington these days. And if you need a decent example of it, look at the child-care policies under the new administration.

You have, in one hand, David Stockman's better basic stick. Any day-care program funded under Title 20 is lumped among the block grants, which have been beated down by 19 percent. Because the states no longer have to put up 25 percent in matching funds, the real blow may amount to 40 percent.

In the other hand, you have the carrots. These particular carrots are growing in the lush vegetable patch of the Reagan tax bill. Here you have some new and really little-known tax incentives that just might feed child-care assistance in the private sector.

The Metzenbaum-Hawkins Amendment to the tax bill served up a couple of helpings to working parents. First, it increased the amount of a child-care tax credit. Then it changed the tax laws in a way that should encourage businesses to offer child-care assistance as a fringe benefit. Here we see the theory: fewer government-funded day-care centers, more government-encouraged private day-care incentives.

As of Jan. 1, a company has more reason to help with that crucial element of worker stability: day care. The company will get not only a tax deduction if it sets up a day-care center on the job, but also a deduction if it (1) pays for employee a day care in other facilities or (2) reimburses workers who pay babysitters in their own homes.

Under the same new rules, a worker will no longer have to report this benefit as taxable income.

Since tax laws run the business world, this could add up to a real boost in industry's day-care assistance for working parents.

As Don Hasbargen of Hewitt Association, a major employee-benefits consulting firm, put it: "Now day care has become a tax-favored transaction. Those companies that would be inclined to provide day-care benefits for employees have an additional reason to think about it."

It's unlikely that companies will suddenly "think about" child care as a standard item on the benefit sheet, like health insurance or a pension plan. There are too many employees who don't use child care and too much sensitivity about unequal benefits.

But it could become a featured item, even a star attraction, in the new flexible packages. These packages, called "cafeteria plans," are only in operation in a dozen major corporations, but they are being considered by many others.

Under a cafeteria plan, a worker can fill a tray of benefits to fit his or her own needs from a line of options, including everything from vacation time to legal insurance to dental care. A working parent with a children, for example, might be able to pick child-care reimbursement rather than, say, vacation days, when the kids are small.

The problem with this transfer program is that the carrot may not feed the same families who are being affected by the stick. The Title 20 programs have been geared to the poor. There will be, at best, a time lag before the effects of the changed tax law are felt. In the short run, there will be a net loss in day-care facilities.

But the new law has a potentially wide and beneficial effect. In the long run, it encourages a diverse system under which parents can create and choose the sort of day care they prefer -- at home, family day care, day-care centers. It encourages industry to consider that other half of workers' lives: home and family.

At best, if this works, it may even protect day care, always fragile in any war for funding, from the ritual beatings of the budget stick.