Members of the government's largest employee union have a double-or-nothing decision to make this summer. The course they take could make them the largest, most influential labor organization in the world or it, could bring a host of legal, administrative and financial hassles that could cripple the union.

The issue for the American Federation of Government Employees is organizing the military. AFGE delegates last year voted to amend the union constrution to permit uniformed military personnel to join. Earlier this year, a task force proposed that the question be put directly to the 27,000-plus white-collar and blue-collar civilians who now make up the AFL-CIO union.

If AFGE locals vote to organize the military, and if large numbers of service personnel take out union cards, membership easily could double or triple. AFGE would have clout in Congress and with the White House that would make it the nation's union pow.

Being able to speak for people in both the civilian and military arm of government would give AFGE leaders new respect with congressional liberals who already tend to go along with improved pay and benefit proposals for government workers. More important, it could open doors with conservatives who openly hate bureaucrats, but who are willing to do handstands for the Pentagon any time military appropriations or benefits are considered.

Organizing the military - the last big bloc of virgin territory in this country for labor unions - could, if successful, give AFGE a big financial shot in the arm. AFGE like other government groups, is hard-pressed because membership dropped then stablized, in recent years. Also the union's dues are only a fraction of dues paid by members in the private sectory, yet demands for legal services, contract negotiations and servicing members grievances are complex and growing.

The negative side of organizing the military is equally awesome, and sobering. It could cost the AFGE a bundle to organize soldiers, sailors and Marines. Both the White House and the Pentagon are opposed to unions in the military, and there has not been what could be called a broad-based groundswell of interest about unions in the military.

Also, many AFGE members - and leaders - fear that the union could lose sight of its principal objective - protecting and improving the lot of civilian government workers - in its bid to sign up uniformed members. There also is the distinct possibility that the union would be flooded with membership applications from military personal whose demands for service and representation would cost far more than their dues-dollars would be worth.

Finally, there is the unknown factor of what an attempt to organize the military would do for the union's image both with the American public and with Congress.

AFGE, leaders have, wisely, taken a relatively middle-of-the-road position. They are leaving the decision to members who will be voting this summer and fall on the military organiztion question.

Meantime, the union has done an excellent job of giving in-depth pro and con agruments to members. The for-and-against debate is in the current issue of the AFGE Standard - the tabloid newspaper that goes to all members. It will probably become "must" reading as well for Congress, too military personnel and ordinary citizens who depend on - and finance - both the civilian and military establishments.

If AFGE members decide to go after the military membership potential, the peacetime military will never be the same. It may be better, or worse, for unionization. But it will never be the same. Neither will the AFGE.

Civil Service Change: The White House has decided to replace veteran CSC member Ludwig J. Andolsek even though he's a Democrat whose term runs through March, 1981. Andolsek, 66, was vicechairman of the merit system agency under Kennedy and Johnson, and minority member during the Nixon and Ford administrations.

His premature (by four years) departure is a sure sign the Carter people will replace minority (Republican) member Georgiana H. Sheldon, too. That would mean CSC would have a brand new team for the first time since 1961.

Former Chairman Robert E. Hampton resigned Inauguration Day after serving with CSC since the Kennedy administration, which also appointed Andolsek. Insiders say the White House concluded - in the wake of merit system abuses revealed during the Nixon years - that it had to start off with new faces at CSC.