Primary responsibility for investigating the growing number of bombings at abortion clinics nationwide belongs to a low-profile federal law enforcement agency that initially made its mark chasing bootleggers and moonshiners.

But supporters of the U.S. Treasury Department's Bureau of Alcohol, Tobacco and Firearms said yesterday that although it is less well-known than the larger, more image-conscious FBI, the ATF is highly regarded within law-enforcement circles for its technical expertise and success in investigating bombing cases.

Since the late 1960s and early 1970s, the ATF has had the primary responsibility for investigating bombings.

The FBI confines its investigations of bombings to select cases, such as instances involving terrorist groups or explosions at federal buildings.

"ATF is probably the premier agency when it comes to explosions and explosive devices. That is their mission," said Rep. William J. Hughes (D-N.J.), chairman of the House Judiciary subcommittee on crime, which has oversight over ATF.

"If you ask local police organizations around this country who they contact for technical expertise and other assistance relating to bombings, more often than not they will tell you ATF," Hughes said.

Supporters of the ATF also said yesterday that the agency has been doing a good job investigating the bombings. Supporters noted that they have made arrests in 12 out of the 30 incidents at abortion clinics that have occurred since 1982, which they said is particularly impressive since most of the incidents have occurred within the past year.

However, critics of the Reagan administration's handling of the investigation, including D.C. Mayor Marion Barry and prochoice groups, have argued that the FBI, as the country's largest and premier law enforcement agency, should be given primary responsibility for coordinating the nationwide investigation.

President Reagan has ordered an all-out federal effort to find those responsible in the incidents at the abortion clinics but has stopped short of directing the FBI to take over the investigation.

Reagan's critics argue that the FBI is a much more important symbol than ATF to most citizens and the failure to give it prime responsibility suggests that the Reagan administration, which supports a constitutional amendment to ban abortions, has not given the investigation top priority.

"I think the FBI psychologically in the minds of the nation has a greater stature in terms of how people view it as a leading arm of law enforcement for the national government," Barry said yesterday.

Rep. Don Edwards (D-Calif.), a former FBI agent and chairman of a House Judiciary subcommittee with oversight over the FBI's domestic security investigations, said yesterday that "you can't compare the expertise and resources the FBI has nationwide with any other federal law enforcement agency."

The FBI has an annual budget of more than $1 billion and employs about 9,000 agents, nearly nine times as many investigators as ATF, which has about 1,200 agents and a budget of about $150 million.

Some local and federal law enforcement officials suggested that the FBI may be reluctant to take the lead in the abortion bombings cases because of the strong criticism it received from Edwards and other congressmen for its spying on left-wing political groups during the civil rights and anti-Vietnam War protests.

FBI officials yesterday said that concern about past criticism of FBI investigations of political groups was not a factor in the decision to have ATF take the lead in the abortion clinic bombings.

William Baker, chief spokesman for FBI Director William Webster, said that under its terrorism guidelines the bureau would become the lead agency in the bombings case if an organized group or conspiracy is behind them. Baker and ATF officials said that the ATF investigation up to this point has found that the bombings are not the work of an organized group.

Nonetheless, Baker said, the FBI has been providing ATF with "every assistance," -- including intelligence and technical information -- "short of having our agents out doing the interviews."