SUNDAY IS Father's Day, and most American children will be doing something special for dear old dad. It's a useful occasion for children to remember that the old man works hard to keep a roof over their heads and that, for all his faults, he's not such a bad fellow to have around the house. But Father's Day will be only an unhappy reminder for millions of children that their fathers no longer care enough about them even to help pay for their upbringing.

The failure of fathers to contribute to support of their children is no longer a problem confined to a substrata of American families. More than 8 million families now lack a male parent, and with 1.2 million new divorces every year, the number continues to grow. Experts estimate that one-half of American children-- from all income levels--will live apart from their fathers for part of their childhood. For the great majority of them, the departure of the father will mean a steep and often permanent drop in their living standards.

Fewer than three of every 10 fatherless families receive regular child support payments from the absent father, and the payments received average less than $2,500 a year. Even when fathers are under court order, less than half pay regularly, and perhaps as many as a third never make a single payment. Contrary to popular belief, many of these delinquent fathers have substantial incomes. A California study showed, moreover, that a year after divorce, while the wife's income typically dropped by 73 percent, the husband's rose by 42 percent.

For most women, pursuing a recalcitrant ex-mate is a bleak and expensive process. Courts have huge backlogs of child-support cases, and even if a judgment is won and arrears are collected, the victory is usually temporary. It is especially easy for fathers to avoid further payments by moving to a different state or, in some cases, even a different county.

In recent years the federal government's Child Support Enforcement program has helped states crack down on absent fathers whose families have been forced onto welfare rolls. The program has already produced significant welfare savings in many states, and the Reagan administration is preparing legislation to strengthen provisions for withholding wages and tax refunds from delinquent parents and helping states coordinate collection efforts. These are sensible proposals. But they do little to help either the families involved--since collections simply offset the typically low welfare benefits--or the equally large numbers of deserted families that have avoided welfare but still scrape by on relatively meager incomes.

As more and more families have become exposed to the weakness of the child-support system, Congress has become increasingly interested in additional measures that would have broader impact. Child support is one issue that appeals--rightly--to all parts of the political spectrum. A prospective welfare saving is only one small part of that concern. A society that cares about its future will make every effort to see that its children are not raised in deprivation and that their parents recognize that the decision to have children entails lifelong responsibilities.