It's a tragedy of the culture: crowds of children growing up financially strapped, because their fathers will not support them. Delinquent dads have condemned millions of children to the welfare rolls and millions more to a painful struggle on the margin of poverty.

In recent years, both the state and federal governments have been growing more efficient in collecting child support from neglectful fathers. But some of the dads have gone to court to fight the new collection methods.

Automatic collection is now the rule in many states. When a father will not pay voluntarily, child support may be withheld from his wages or his state income-tax refunds. The Reagan administration wants to make this mandatory in all states.

Two years ago, the IRS began diverting federal income-tax refunds to help pay child support, if the father was seriously delinquent and his children were on welfare. And starting last year, states were required to withhold child support from unemployment benefits, in some cases. (A few runaway mothers come in for the same treatment, but most of the problem is with fathers.)

When a family is on welfare, money recovered from the father is used to reimburse the state treasury. All but 18 states or territories last year recouped more money for the public purse than they spent in tracking down fathers and forcing them to pay. So the program can actually add money to state budgets. (Non-welfare mothers get to keep whatever money the state can collect for them.)

But more than two-thirds of the money for state child-support collection programs comes from the federal government, and last year the Reagan administration reduced the federal share of the cost. This was expected to force state offices to cut back, resulting in a $90 million saving on the federal level. It might also have reduced the number of delinquent fathers caught.

Happily, state-level cutbacks did not come quite as fast as Washington expected. "Our state chose to come up with the money to make up for the federal loss," says Jerrold Brockmyre, director of Michigan's office of child support. Duane Campbell, director of the office of child support in Wisconsin, says: "There is a lot of support from the governor for this program, so it posed no major problem to get additional funds."

But in Kentucky, where the legislature has not gone out of its way to assist the child-support office, director Hanson Williams expects some counties to decide to drop out of the enforcement program this year. Kentucky's program is inefficient; it is not known whether the state will come up with the money needed to keep collections going at last year's levels.

Starting next year, the administration proposes to save more money by changing the way it calculates federal payments to state programs. State directors are wary of the change. They're afraid it will cut their federal funds and set back their collection efforts. Brockmyre says the new payment formula may encourage states to spend more time on welfare collections and give less help to strapped mothers who are not on welfare.

John Gunther, in the federal program's budget office, thinks the states are crying wolf. He says that 40 states should do as well or better with the new form of payment, and that bonuses will be given for helping non-welfare mothers. The change was designed to goad the inefficient states, which he says should be able to regain full federal payments with just modest improvements in their programs.

Gunther had better be right. If he's not, one of the government's most valuable and most sensible programs will be bruised. here are other threats to the national effort to collect child support. Groups of delinquent fathers (whose children are on welfare) are bringing lawsuits to try to stop the IRS from intercepting their income-tax refunds. A Connecticut lawsuit forced the state to change its procedures, but fortunately didn't delay the program.

Mothers who are owed child support, and who can't locate their children's fathers or haven't got the money to sue, should ask the state office of child-support enforcement for help. It is usually located in the welfare department. In some states, non-welfare mothers have to pay a fee for the service. But it's worth it, if the office can find the delinquent dad and make him pay.