Prime Minister Menachem Begin proposed today that Israel, which has executed only Nazi war criminal Adolf Eichmann in its 30-year-history, impose the death penalty for extreme terrorism similar to Sunday's Palestinian attack on Nahariya.

After hanging Eichmann in 1961 for his role in the mass murder of European Jewry by Nazi Germany, the Israeli government decided in 1967 to suspend capital punishment. Begin told a parliamentary committee that he will ask his Cabinet on Sunday to end that suspension, but to apply the penalty only for "extremely cruel acts" of terrorism.

The death penalty proposal coincided with Israeli air raids in Lebanon and a government confirmation that Israeli gunships for the last two days have been shelling Palestinian guerrilla strongholds in the vicinity of guerrilla strongholds in the vicinity of Tyre, a coastal town in southern Lebanon.

Both developments underscored Israeli outrage at the Palestinian raid Sunday, in which four Israelis, including two children, lost their lives.

The gunboat shelling had been reported previously from Beirut, but today's announcement was the first Israeli government acknowledgment that its retaliatory strikes were continuing regularly.

The military command said the air strikes were directed against Palestinian artillery positions near the crusader-era Beaufort Castle and the towns of Artoun and Nabatiyeh. Reports from Beirut quoted local residents as saying six persons were killed in the naval bombardment and three others in the air raids.

[The official Palestinian news agency, Wafa, reported antiaircraft batteries shot down one of Israel's F4 Phantom jets near Tyre, 14 miles north of the border, according to United Press International. The Israeli command said all its planes returned safely.]

"The policy is for the Israeli Defense Forces to take action against terrorist organizations at any time and any place it finds necessary," said an Israeli Army communique.

Defense sources said today's gunboat shelling scored "good hits" on operating bases used in launching terrorist attacks against Israel. Reports from Lebanon said hundreds of residents have been leaving the target area.

Begin, who said at yesterday's funeral for the victims of the Nahariya attack that "not even the devil has yet invented the proper revenge for the murder of an innocent child," is not expected to encounter significant opposition to his request for reinstatement of the death penalty.

He was referring to the death of Einat Haran, 5, who was slain with her father, Danny, after the two were taken hostage from their apartment by a terrorist squad that had landed nearby in a rubber boat. Another daughter, Yael, 2, is believed to have been smothered accidentally by her mother, Semadar, in an effort to keep her quiet while hiding from the terrorists.

Deputy Prime Minister Yigael Yadin endorsed the death penalty proposal, but said he wanted it broadened to cover any heinous criminal acts, not only terrorism. Avraham Sharir, chairman of the Likud coalition, proposed even before Begin's announcement that the death penalty be applied to murderers of children.

Meanwhile, it was announced that Defense Minister Ezer Weizman will go to Cairo on Wednesday to discuss technical aspects of Israeli withdrawal from teh Sinai Peninsula. The trip has been postponed as a result of the Nahariya atack. Army Chief of Staff Rafael Eitan, who had been scheduled to travel with Weizman, will stay behind.

Also Wednesday, Israeli officials led by Eliahu Ben-Elissar, director-general of the prime minister's office, will travel to a U.S. manned listening post in the Sinai buffer zone to exchange documents with Egyptian officials attesting that the Egyptian-Israeli peace treaty has been ratified by the parliaments of both countries.