The Democratic National Committee concerned about dissent in party ranks, is sending out a new message to Democratic candidates across the country: stop taking "cheap shots" at President Carter.

Democratic national chairman John C. White said yesterday he was "distressed" with the degrading comments an increasing number of Democratic officeholders and candidates are making about Carter, and offered them "a little friendly advice" to lay off the president.

"Personal attacks on members of your own party is poor politics," White told a breakfast meeting of reporters, adding later, "the president is a sitting duck" for such criticism. "He can't do anything about it.

"You don't see the real hitters in either party going around and criticizing fellow party members," the Democratic chairman said.

White refused to name the Democrats he thinks are guilty of taking "cheap shots" at the president.

But he is known to be extremely upset with statements by Democratic senatorial nominees Bill Bradley (New Jersey) and Andrew Miller (Virginia) and incumbent Sen. Floyd K. Haskell (D-Colo.) that they consider Carter a liability and would rather not have him appear in their states this fall.

Candidates "would be better off to run on their own merits and records" than against a president in their own party, he added.

The Democratic Party, White said, "encourages dissent and disagreement" on issues, but considers personal attacks on the president out of bounds.

White attributed the attacks on Carter to a short-sighted desire for publicity. "Candidates get awfully dry and they get tired of nobody listening to them," he said. "But when they critize the president everyone stands up and listens to them. Every county road supervisor in the country can get a line in the paper by criticizing the president."

White also said he expects Carter to be challenged in the Democratic primaries in 1980, but not by any major candidates such as California Gov. Edmund G. (Jerry) Brown Jr. or Sen. Edward Kennedy (D-Mass).