From a corny media stunt in which House Republicans waved baseball bats on the Capitol lawn to letters and telephone calls to campaign contributors to conservative Democrats, Republicans are playing hardball in an effort to win passage of President Reagan's budget proposals.

The baseball bats and gold lapel pins depicting crossed bats are part of a multimillion-dollar Republican response to a remark by House Speaker Thomas P. (Tip) O'Neil Jr. (D-Mass.), who told the Presdient last month that he should be prepared "to play hardball" when pushing his legislative program.

A three-pronged attack -- mounted by the White House, the Republican National Committee and congressional Republicans -- is aimed at convincing at least 26 of 62 targeted Democrates to join House Republicans in supporting an alternative to the Democratic budget proposed by Budget Committee Chairman James R. Jones ; (D-Okla.).

During Congress' Easter Recess, the RNC dispatched two dozen officials, including Vice President Bush, Office of Management and Budget Director David A. Stockman, former president Gerald R. Ford and a score of congressmen on a flying trip to the South, where they spoke in the home districts of some of the 62 targeted Democratic House members from whom the GOP hopes to get enough defectors to overcome the Democratic majority and pass the Reagan plan.

At the same time, the RNC pulled from public records of the Federal Election Commission the names of persons and organizations who gave money to both Reagan and conservative Democratic congressmen, and implored them to contact their congressman.The recuperating president was among the callers.

RNC chairman Richard Richards, in a letter to 200,000 GOP contributors, urged recipients to send an enclosed postcard to O'Neill urging him "without delay to pass the complete Reagan economic package."

O'Neill aide Gary Hymel said wastebaskets in the speaker's office are overflowing with 20,000 of those postcards, from which five were saved because they contained kind words, and small checks, for a Democratic rebuttal.

Down the street from the Capitol, 43,000 letters responded to a section of Richards' letter that asked for contributions to an $11.5 million war chest for 1981, of which about $2.5 million will be spent on an advertising campaign to support the Reagan program.

Other facets of the plan include sending letters to the editor and "op ed" pieces to hometown newspapers of conservative Democrats; "Dear Colleague" letters to vulnerable Democrats, and a barrage of special publications, including 900,000 copies of a full-color special editon of First Monday, the RNC monthly magazine.

Evidence abounds that the blitz is working. Even before the president's televised speech Tuesday, O'Neill conceded that 28 Democrats were supposed to have signed on with the President, although the speaker said he had wooed three of them back, and "more could be well be returning."

Rep. Stanford E. Parris (R-Va.), chairman of the House Task Force on Economic Policy, a GOP group, said the key to acceptance of the president's program is convincing members that the folks back home are looking for a clear-cut up-or-down vote from them. t

"No one is in agreement with every single item," Parris told Rep. Roy Dyson (D-Md.) during a visit to Dyson's office this week as another phase of the all-out attack.

Armed with the knowledge that Reagan outpolled Dyson in Maryland's 1st Congressional District and that Dyson may face a tough reelection campaign next year from Republican former representative Robert E. Bauman, Parris told the freshman Democrat that "your district is as conservative as any I know."

The upcoming vote is a chance to "make a broad policy statement" without binding a member to specific cuts, Paris said.

Parris left without a commitment from Dyson, who said he also had been called by a White House aide, but Dyson accepted a crossed-bats lapel pin, saying he would wear it if he decides to go along with the president.