President Carter's Christmas cards are being mailed this year at the expense of the Democratic National Committee. With each card goes a silent prayer that the recipient will be softened up enough by the gesture to help Carter's embattled legislative program and his campaign for re-election.

The "Christmas card" strategy was outlined in a memo to the President last month from Paul Sullivan. Democratic Party executive director. He noted that there were thousands of past Carter supporters who might be mobilized as part-time lobbyists for his 1978 program and his 1980 campaign, but who show some "disaffection" because they "have received nothing from the Carter campaign since the election - no acknowledgement of their efforts, no thank you for their work."

In his "scenario" for reactivating the Carter volunteers, which presidential assistant Hamilton Jordan said yesterday has won a "favorable" White House reaction. Sullivan said the first step should be "inclusion of this entire pool in the Christmas card mailing which will be underwritten by the DNC."

"Not only will this provide for first contact which is not a solicitation and let them know that low-(sic) and behold someone does remember them, but it can also serve to purge our list of incorrect addresses and get new ones," Sullivan wrote.

White House officials said that the first 30,000 cards from the President and Mrs. Carter - featuring a pen-and-ink drawing of the White House by Georgia artist Harvey Moriarty - were mailed yesterday, after crews of volunteers had worked for two weeks addressing and stuffing the envelopes.

Sullivan said it cost the DNC about 20 cents apiece to buy the cards from the Hallmark Co. and mail them. Neither he nor White House officials could say how many more cards would be mailed.

Sullivan said in the memo to the President that the DNC had collected about 150,000 names of past Carter supporters it hopes to solicit next month for the proposed "Support-Our-President" organization. Each member would be asked to contribute $10 every six months to finance its activities.

Sullivan said the organization could be used to:

"Generate mail to Congress on behalf of the President's program.

"Generate mail or phone calls to state legislators such as on behalf of the Equal Rights Amendment.

"Do volunteer work at the President's request for a congressional candidate.

"Identify supporters to run for delegate to the National Party Conference," to be held next December.

"Generate letters-to-the-editor and commentaries from local people . . . and provide well-informed articulate supporters of the President's policies to participate in panel discussions . . . debates and radio call-in programs.

"Provide a ready-made organization for the 1960 primaries and general election."

A key feature of the proposed organization is a phone network or "tree," designed to alert the volunteers. Sullivan outlined a series of steps - some public and some concealed from the volunteers - which he said would "reduce the possibility of someone sabotaging our organization."

The performance records of the volunteers would be entered on a DNC computer which he said "will be programmed to monitor the reliability of each participant . . . and identify weak links for replacement or shoring up."

"At approximately six-month intervals," he wrote, "we will re-form our (telephone networks) tree - promoting or demoting people, based on reliability - using the computer."

But Sullivan indicated the whole project might be stymied by the disaffection of those past Carter supporters who feel they have been neglected since the election.

He reported that DNC interns who tried last summer to obtain names of Carter workers were stymied in some states by "dissatisfaction over patronage."

While most of the 1976 Carter volunteers "certainly did not expect a job or any special privileges," Sullivan said, "people appreciate being thanked."

He told the President in the memo: "We need to develop a mailing approach which will minimize disaffection, i.e., 'I worked for a year for Jimmy Carter and wasn't even thanked. Now thy want me to work for him again. Why should I?"

The answer he suggested - and which was adopted - was the Christmas card strategy.