Jeane J. Kirkpatrick, a Hubert Humphrey Democrat who found happiness in President Reagan's Cabinet, yesterday completed her conversion to the Republican Party, saying she is "tired of swimming against the current in my own party."

Sensing a public relations bonanza in Kirkpatrick's long-expected switch, the Republicans welcomed her officially at a big fund-raising reception for the newly formed GOP Women's Political Action League (GOPAL) and touted her as a symbol of a realignment they said was under way in American politics.

Reagan sent a letter in which he noted that he was a "convert myself," and told Kirkpatrick that the "water's fine on this side of the dam."

The letter was read by Vice President Bush, who said Kirkpatrick's conversion "proves our party has no limits" and called Kirkpatrick "already, deservedly so, one of most celebrated members of the Republican Party."

Kirkpatrick, who recently stepped down as Reagan's United Nations ambassador and reregistered yesterday as a Republican in Montgomery County, did not back away from such praise.

She said she had changed parties because the Democrats no longer stood for "growth at home, strength abroad."

"I think those positions effectively are more clearly represented today in the dominant portion of the Republican Party than in the dominant portion of the Democratic Party," she said at a news conference yesterday afternoon.

"I don't think the policy positions of the dominant wing of the Democratic Party on foreign affairs make much sense today, quite frankly," she added. "I think the Republican Party makes a lot more sense, and that's a very important measure of why I'm here."

Wishing her friends in the Democratic Party "good luck" in trying to reshape that institution, she said that although she does not agree with everything Republicans stand for, she is "very sympatico" with her new party. "I intend to swim with that current," she said. "I'm tired of swimming against the current in my own party."

Democrats took a jab at Kirkpatrick's switch. "At a time when the Republican Party of Sen. Jesse Helms and Jerry Falwell is moving to the right-wing fringes and the Democratic Party is reaffirming its historic commitment to mainstream policies of economic growth and a sound, efficient national defense, it is difficult to understand why Jeane Kirkpatrick is becoming a Republican," said Terry Michael, a spokesman for the Democratic National Committee.

Kirkpatrick's hard-line stance on foreign policy and her sharp attacks on the Democrats have made her a favorite of GOP conservatives, who have touted her as a possible vice-presidential candidate in 1988.

She was joined at yesterday's news conference by Republican National Chairman Frank J. Fahrenkopf Jr. and Maureen Reagan, the president's daughter and a founder of GOPAL.

Asked about her interest in running for office, Kirkpatrick, the most prominent Republican convert since former Texas governor John B. Connally in 1973, referred to statements she made a decade ago after publication of one of her books. "I am more interested in reading and writing and thinking about politics, and that's my position today," she said.

Asked if that meant she was ruling out running for office, she replied: "You know what I said in the U.N. for four years . . . ? I said what I meant and I meant what I said."

Kirkpatrick recently ruled out challenging Sen. Charles McC. Mathias Jr. (R-Md.) in a GOP primary next year.

The newest Republican is a veteran of Democratic Party politics and a disciple of the Humphrey and Henry M. Jackson school of internationalism in foreign policy and big government at home. For years she was part of the party wing that resisted what she described yesterday as a "neoisolationist" stance in reaction to America's defeat in Vietnam and such domestic policy "extremes" as quotas to achieve racial balance.

She said she believes that a political realignment in America "is very far advanced" and will overturn the dominance the Democrats have enjoyed since Franklin D. Roosevelt was elected president in 1932.

"The Roosevelt coalition was a response to a set of desperately serious problems, crises in American life," she said. "I think the Roosevelt coalition provided a response that was valid, important, humane and absolutely, vitally necessary to our nation. I think the Reagan coalition . . . provides responses to contemporary problems which are as relevant and as important."