Senate Democratic leader Robert Byrd said Thursday President Reagan should fire Attorney General Edwin Meese on grounds the operation of the Justice Department is faltering because of his chronic legal problems.

Meese has rejected stepping down, and, after a visit to Capitol Hill Wednesday, apparently is ready to fight through "the last days" of Reagan's term, despite harsher criticism within his own party.As he left the Justice Department Wednesday to deliver a speech, Meese hinted he fully expects to be attorney general until Reagan leaves office in January.

"We are going on the offensive," he declared. "We have a very vigorous program in the Department of Justice. . . . The battle against drugs, continuing our efforts against obscenity and organized crime, what we're doing to combat terrorism, all of these things are going to be vigorously pursued right down to the last days of this administration."

Byrd, D-W.Va., said that whatever the validity of the allegations against Meese, the Department of Justice is not operating efficiently "because he is spending an inordinate amount of his time dealing with his own personal problems.