John B. Anderson, the former Republican Illinois congressman who ran for president as an independent last year, has begun the first cautious exploration of a new drive for the presidency as the head of a third party in 1984.

Anderson's supporters last week filed papers at the Federal Election Commission forming the National Unity Committee, an outgrowth of Anderson's 1980 independent "national unity" presidential campaign organization, which is to be the primary vehicle for his continued political activity.

Next week, a fundraising letter from Anderson asking for contributions to the committee will be mailed to 50,000 known Anderson supporters, according to Jane Fowler, Anderson's executive assistant in his Washington office.

Fowler said that if the first fundraising letter is successful it will be followed by a second mailing to some 175,000 contributors to the 1980 Anderson presidential campaign.

Fowler said a decision by Anderson to form a third political party for the 1984 presidential election will depend on the response to the National Unity Committee's activities and is not likely to be made until after the 1982 congressional elections.

But she said Anderson supporters around the country plan to field slates of third-party candidates in local, state and congressional elections next year in at least six states, including Michigan, Ohio, Illinois and Minnesota.

In next week's first fundraising letter, Anderson makes clear he hopes to use the newly formed committee to revitalize and expand the independent campaign organization that produced 5.7 million votes, or 6.6 percent of the total, for him in 1980.

Contributions to the committee, he says in the letter, will be used to maintain a small Washington office staff to perform logistical, organizational and speech-writing functions, to produce a bimonthly newsletter and to make a state-by-state survey of the organization's political strength.

"Through the newsletter and our meetings with you, state by state, region by region, the decision before us will become apparent -- whether to continue our efforts or leave our fate to the existing parties," Anderson says.

Anderson also makes it clear that he believes he and his supporters should have "a continuing voice in the political processes of our country." He says President Reagan's economic goals are "laudable," but he predicts the tax cuts sought by the administration will only fuel inflation, and says Reagan's "draconian reductions in funding for proven 'people programs' . . . are incompatible with the concept of a progressive, caring society."

The 20-year House veteran, who is teaching at Stanford University and the University of Illinois as well as doing commentaries for a Chicago television station, also uses the fundraising letter to criticize Reagan on energy issues, human rights and foreign policy in Central America.

If the new president fails, the only alternative should not be the Democrats, Anderson suggests. His supporters should explore whether there are "new political alternatives which can attract the support of the increasingly frustrated and alienated American people."