WASHINGTON (AP) _ Democratic Sen. Joseph R. Biden Jr. and Republican Sen. Bob Dole are hammering their organizations into shape for the grueling presidential nomination campaigns, as each seeks to benefit from the departures of well- known hopefuls from the pool of 1988 contenders.

Each man took a step toward an official candidacy Tuesday with the formal establishment of campaign committees.

Biden, the Senate Judiciary Committee chairman, filed papers with the Federal Election Commission setting up a formal ''Biden for President'' campaign committee.

Dole, the Senate minority leader from Kansas, did not go quite as far as Biden, but he did form an exploratory committee Tuesday. Such an organization is not a formal campaign committee required to file reports to the FEC, but it is usually converted to the full-scale committee as the candidate moves toward a formal announcement.

The next step for Biden, a three-term senator from Delaware, is expected in late May or early June, when he has said he will formally announce his candidacy.

There is no timetable for Dole to make his candidacy for the Republican nomination official, Robert F. Ellsworth, chairman of the Dole committee, told a news conference.

Dole is working to tighten his grip on the No. 2 spot in the Republican race by picking up the support of former backers of Howard H. Baker Jr. Baker, Dole's predecessor as Senate GOP leader, dropped out the GOP race when he agreed to become President Reagan's new chief of staff last Friday.

''We have been in touch with a number of Baker supporters,'' Ellsworth said.

Chief among those Baker backers Dole is seeking to recruit is Republican Sen. Warren Rudman of New Hampshire, the state with the nation's first presidential primary.

Dole has also benefited from the fallout of the Iran-Contra controversy, which has damaged the poll standings of front-runner Vice President George Bush.

''The Iran-Contra affair has tended to focus favorable attention on Dole,'' Ellsworth said. ''How to measure that, I don't know.''

On the Democratic side, Biden is one of the beneficiaries of the declaration by New York Gov. Mario Cuomo last month that he would not enter the 1988 Democratic race. Cuomo was running second in the polls to Democratic front-runner Gary Hart, with Biden and all the other hopefuls far behind.

With no clear second choice in the Democratic ranks, Biden, an impressive speaker, is given a shot at becoming the main alternative to Hart. In addition, Democratic party sources, who spoke on the condition of anonymity, said Biden has close ties to big Democratic fund-raisers in New York who are now free to work for him instead of Cuomo.

Also on Tuesday, both presidential hopefuls filled key spots in the campaign organization.

Biden named his sister and longtime campaign manager, Valerie Biden Owens, as the manager of his presidential effort. He also announced that David Wilhelm, an official with the AFL-CIO in Washington, will be his Iowa campaign manager.

Ellsworth, a former Kansas congressman and ambassador to NATO, was formally named chairman of Dole's effort Tuesday. He said the campaign was negotiating with David A. Keene, chairman of the American Conservative Union, about a role in the campaign. That left uncertain what role, if any, John Sears, would have.

Keene has said he would not work in the Dole campaign if Sears was the manager. Sears was Ronald Reagan's campaign manager in 1976 and early 1980. He was fired the day of the 1980 New Hampshire primary.

Former Secretary of State Alexander M. Haig Jr. plans to enter the Republican race March 23, and Rep. Jack Kemp of New York will join the field on April 6. Bush has an exploratory committee, but aides say he won't declare his candidacy until next fall.

Biden joins two other Democratic hopefuls who have formal campaign committees. One, Rep. Richard Gephardt of Missouri, has also formally announced his candidacy. The other, former Arizona Gov. Bruce Babbitt, set up a campaign committee some time ago and will announce his candidacy formally on March 10.

Hart has set up an exploratory committee, and will announce his second bid for the Democratic nomination on April 13.