THE AIR traffic controllers struck for much more than money. The union's own leadership misunderstood that point when it negotiated the conventional wage compromise that its members, in a fury, rejected. By walking off their jobs, they put President Reagan in an impossible position, he has been right to uphold the law rigorously and end the affair quickly by firing the strikers. But, as the administration begins to rebuild the corps of controllers, the administration needs to think carefully about the causes of this unhappy episode. The gigantic claims for pay increases were hardly more than surrogates for other, and more deeply felt, claims.

There is a certain style of management that deals with lower-echelon complaints by arranging never to hear about them.The word is passed down the ladder to shut up and get on with the job. It's pretty clear, not merely from the strikers but from congressional inquiry before the strike, that the Federal Aviation Administration has often managed the air control system in that spirit.

The FAA operates, for example, highly sophisticated networks of radars and computers to ensure air safety. But like all highly sophisticated electronic networks, they occasionally go down and leave the controllers to cope as best they can. Controllers seem to believe that they will be held personally responsible for any accidents during these lapses. The people at the top of the FAA, trying to reassure Congress and the public about the safety of air travel, resort to methods of reporting that, to the controllers, seem to understate grossly the frequency and extent of the system's failures. Complaints travel down the chain of command but not, apparently, up.

A degree of stress is inherent in the controllers' work, but no more than in many other kinds of jobs. Unfortunately, it's evident that this stress has been severely compounded by other tensions and frustrations generated less by the nature of the work than by the nature of the FAA.

This strike was an indictment of a union that allowed its members to rush into a foredoomed venture with so little sense of the consequences. But the strike also raises sharp questions about the way the FAA is being run. There is not much point in recruiting and training a new generation of controllers merely to repeat the errors of management and morale than came to a culmination this summer.

The secretary of transportation, Drew Lewis, has skillfully surmounted the challenge of the strike. But the next part of the job requires more than good tactics. It requires a willingness to examine closely the strikers' grievances and ensure that there will be little ground for them among the new recruits.