New federalism and the balanced-budget amendment dominated the public discussion at the National Governors Association meeting that ended here today, but the prospect of Republican statehouse losses in the November elections fueled the political talk.

Thirty-six states will hold gubernatorial elections this fall. Democrats now hold 20 of those statehouses, the Republicans 16. But the Republicans are expected to lose some of their seats because of the recession and high unemployment in many states and the retirements of several incumbents.

Sensing that the statehouses may provide them with their greatest opportunities this fall, Democrats have formed a Committee to Elect Democratic Governors and plan their most extensive effort ever on gubernatorial elections.

Democratic national chairman Charles T. Manatt, who attended the sessions here at a resort called Shangri-La, predicted his party will gain four or five governorships in November.

The Republicans also are fighting history. Every time they have won the White House in this century, they have suffered substantial gubernatorial losses in the midterm elections that followed. In 1922, they lost 12 governorships; in 1954, they lost 8; and in 1970, they lost 11.

Conversations with politicians and officials from both parties here this week indicated that only three of the Republican governors running for reelection now appear safe. They are Pennsylvania's Richard L. Thornburgh, South Dakota's William Janklow and Tennessee's Lamar Alexander.

Alexander, however, may face a serious challenge from Knoxville Mayor Randy Tyree, who won an impressive nomination victory last week and will be well-financed this fall.

The Republicans are most vulnerable in the Midwest, now their area of greatest strength, where the recession has devastated the states' economies. In Michigan, the worst of the lot, the unemployment rate in July was 14.4 percent.

Complicating the Republicans' problem there is the fact that five incumbent governors are not seeking reelection: Minnesota's Al Quie, who was in serious trouble; Iowa's popular Gov. Robert D. Ray; Michigan's William G. Milliken; Wisconsin's Lee S. Dreyfus; and Ohio's James A. Rhodes, who is ineligible to run again.

The lone Republican incumbent running in the Midwest, Illinois Gov. James R. Thompson, is in trouble because of his acceptance of expensive antiques and paintings from supporters and his use of campaign contributions for babysitting.

In Iowa, however, Republican prospects have improved since Democratic nominee Roxanne Conlin admitted she and her husband had taken advantage of tax shelters to avoid payment of state income taxes and reduce their federal tax liability.

The economy also threatens the political survival of Oregon Gov. Victor G. Atiyeh (R).

Two other Republicans, both elected in surprise victories, face difficult reelection races this fall.

Texas Gov. Bill Clements Jr., the first Republican chief executive in that state in a century, is favored to defeat Democratic Attorney General Mark A. White. Not even Republicans believe it will be easy, however, because of the historic strength of the Democratic Party and Clement's brusque style that has offended many Texans.

In Arkansas, Republican Frank White is in a rematch with former Democratic governor Bill Clinton, the most surprising loser in 1980 gubernatorial campaigns. In a state that is overwhelmingly Democratic, White is a clear underdog.

Republicans now believe they may be able to win the governorship of California, where incumbent Gov. Edmund G. (Jerry) Brown Jr. is running for the Senate.

Earlier polls had showed Los Angeles Mayor Tom Bradley with a strong lead over potential Republican candidates, but today, the Republican nominee, Attorney General George Deukmejian, is given a shot at beating Bradley, in part because he is well-positioned on the issue of crime.

Republicans here also expressed hope of winning the governorship of New York, where incumbent Gov. Hugh L. Carey is retiring.

In the Plains states, Republicans hope to defeat Kansas Gov. John Carlin, while Democrats have their eye on Gov. Charles Thone of Nebraska.

The greatest concentration of Democratic governors today is in the mountain West, a region that Reagan swept in 1980 and that has given the Senate many of its most conservative Republicans.

Despite Reagan's continuing popularity there, most Democratic governors running for reelection appear safe, while the lone Republican, Nevada's Robert List, is in trouble because of the MX missile and his decision to raise the state sales tax.

Among the Democrats, only Idaho's John V. Evans appears in trouble today, principally because of economic problems in his state. Arizona's Bruce Babbitt and Colorado's Richard D. Lamm are running strongly and Wyoming's Ed Herschler is expected to win reelection. In New Mexico, where incumbent Democrat Bruce King cannot succeed himself, the Democratic nominee Toney Anaya is the favorite.

In addition to Idaho's Evans and Kansas's Carlin, the other Democratic governor considered vulnerable today is Hawaii's George R. Ariyoshi.