Washington — Republicans are staying cool. In the wake of the historymaking selection of Geraldine A. Ferraro as Walter F. Mondale's running mate on the Democratic presidential ticket, Republican Party leaders indicate they will have a two-pronged strategy on the hustings:
* They will compare the qualifications and experience of the vice-presidential candidates. Geraldine Ferraro, they say, cannot begin to measure up to incumbent George Bush, who has a long record of distinguished service in the foreign policy field and the experience of incumbency.
* They will get out the message that President Reagan and the Republican Party have done more for women than the Democrats.
''As a woman I congratulate Geraldine Ferraro,'' says Betty Heitman, cochairman of the Republican National Committee. ''It's a great step forward and she will be breaking new ground and opening the way for other women. But when the American public looks at who will be heartbeat away from the presidency . . . they will be looking at qualifications and experience and at the record of what the President has done.''
''There is an excitement on the Democratic side, and Mondale will come out of the convention better off than when he went in,'' says James Lake, communications director of the Reagan-Bush Campaign Committee. ''But after that rosy glow and people begin to look at what they want of a vice-president of the US - his qualifications and leadership qualities - we will do very well.''
In a quick sampling around the nation, Republican state chairmen say that the Ferraro choice poses no new challenge to GOP election chances, although they do not deny that, in some areas, it will draw women's votes away from President Reagan.
''I'm relieved that it's not a Texan and not Cisneros,'' says George W. Strake, chairman of the Republican State Committee in Texas, a key swing state in the coming election. ''When the dust settles, we'll be pointing out that it's a popular Reagan-Bush ticket running against two Northern liberals, and that will help the Phil Graham race (for the US Senate seat being vacated by GOP Senator John Tower) as well.
''It appears that Mondale is playing to his strength in the labor-dominated states and trying to get some ethnic support in the Eastern seaboard states. He's probably written off the South.''
''It's not the gender. It's the background and experience, and people will choose Bush over Ferraro.''
In Pennsylvania, another important battleground, Robert B. Asher, chairman of the GOP state committee, says ''you're not going to fool women by just running a woman.
''It will not have that much impact on the race,'' adds Mr. Asher.
''I don't believe there is a gender gap,'' he says. ''The President's record on women and women appointments is outstanding. The Ferraro selection will not change our strategy.''
The Reagan campaign has already mounted a vigorous drive to court women's votes and close the so-called gender gap. Women-for-Reagan committees have been formed in the states. A National Women's Coalition has been created by the Republican Party to enlist women business executives and other professionals in the Reagan campaign. The President's daughter, Maureen Reagan, is central to the entire effort to persuade American women that their interests are best served by the Republican Party.
Mrs. Heitman says, historically, it is the Republicans who have trained women in political skills and given them leadership roles.
In 1976, she recalls, Anne Armstrong ''came close'' to getting the vice-presidential nomination.
Republicans also point to the number of women appointed to high office by President Reagan, including Supreme Court Justice Sandra Day O'Connor.
Democrats see the matter quite differently, especially in Midwestern states where the election is expected to be closely fought.
''Indiana is a difficult state for Democrats,'' says Indiana State Sen. Julia M. Carson. ''Ferraro will be an asset to the ticket and (will have) an appeal in our state.There are many women in the work force, and they have suffered.''
But Indiana GOP State Committee chairman Gordon K. Durnil disagrees. ''Sex will not make a difference,'' he says. ''The Democrats are running a female for lieutenant governor, and many voters will vote against the governor because he selected a woman. The key is name recognition and qualifications, and our surveys show Reagan running ahead of Mondale 62 to 33.''
Ohio Democratic state committee member Helen Karpinski says the choice of Ferraro will have ''a great impact'' in her state. She notes that when Democratic women ran in county races for judicial posts in strongly Republican areas they won. ''This could mean a swing of Ohio to Mondale,'' she says.
''It won't change my strategy,'' says New York state GOP Committee member Eunice B. Whittlesey.
''The Democrats have promoted and promoted the gender gap, ignoring the facts. It is incumbent on us in the party to get the facts out and put out the positive things we have done,'' she says.
''In New York this will gain some votes for Democrats and it will lose some. If people perceive that this is is a gimmick, it's unlikely to pick up votes. There are people who will perceive it as tokenism because it's not the most important slot in the world.''