The apparently crucial part played by Syria in resolving the Beirut hostage crisis has raised the prospect of a whole new U.S. strategy for dealing with terrorism in the Middle East, and for improving relations with President Hafez Assad, long regarded as an adamant foe of America.

Assad has now clearly demonstrated his willingness to be "helpful" to the United States, an idea first suggested by Assistant Secretary of State Richard W. Murphy almost a year ago. His suggestion raised a storm of protest in Congress at the time because of Syria's leading role in driving the United States out of Lebanon.

It is too early to say whether the Reagan administration, in response to Syrian help in ending the Beirut hostage crisis, has any plans for forging a closer relationship with Assad. National security affairs adviser Robert C. McFarlane said last night on Cable News Network that Syria and the United States were still in "considerable disagreement" over the Middle East peace process but that the issue of terrorism was of global importance and "to the extent Syria can help, and has, we welcome that."

But a solid rationale has now emerged for the first time for cooperation between Syria and the United States -- namely, a common perception of the need for a crackdown on uncontrolled Shiite Moslem extremism that threatens the interests of both countries.

There may even be a basis for a three-way agreement on this need that could include Israel, which faces the specter of Shiites belonging to Hezbollah, or "Party of God," displacing the less fanatical Amal militia in southern Lebanon along Israel's border.

Hezbollah's forces represent a grave challenge to Syria's own geopolitical role, particularly its ability to control events in neighboring Lebanon, which Assad considers a virtual province of Syria.

Hezbollah, with this hijacking, confirmed that it is also a threat to the United States. The United States has long been concerned about the ability of radical Shiites -- Hezbollah and others -- to endanger the conservative Arab states that are crucial to American interests in the Middle East.

Should Syria clamp down, and there are reports from Beirut and Damascus suggesting it fully intends to, then Damascus could well prove an invaluable ally in the U.S. campaign against terrorism -- an ironic twist in its own past strategy of encouraging, and even using, the extremists to achieve Syrian ends against the United States in Lebanon.

A crackdown by Syria and its main Shiite ally in Lebanon, Nabih Berri's Amal militia, on the extremist Iranian-backed Hezbollah would give the Reagan administration a convenient diplomatic alternative to launching its own retaliatory strike on this shadowy group, an approach that carries grave risks for U.S. interests in the Middle East.

Furthermore, given Syria's record for dealing ruthlessly with those Damascus perceives as opposing its will, a Syrian-backed crackdown would certainly be far more effective than any brief military action by the United States.

Unconfirmed Lebanese reports are already circulating in Washington that the administration and Syria have discussed the possibility of Syrian forces' taking control of Beirut airport security with American blessings. This could be an alternative to one option for retaliation already threatened by the Reagan administration: shutting down that airport, which U.S. officials regard as one of the world's last havens for hijackers, by either military means or an international boycott.

There is also speculation that the United States could help mediate an agreement between Israel and the Syrian-backed Amal to give Amal a stronger role in southern Lebanon at Hezbollah's expense. In recent weeks, Hezbollah forces have demonstrated increasing influence in that area, so sensitive to Israel.

One of the more interesting developments was evidence that Syria had used its considerable influence with Iran to pressure the hijackers and their Hezbollah backers, according to U.S. officials. Syria has been one of only two Arab countries to support Iran against Iraq; it has given Iran's Revolutionary Guard access to eastern Lebanon, where it maintains bases and helps train Hezbollah members.

Last Sunday and Monday, in the midst of the crisis, Assad met in Damascus with a high-level Iranian delegation led by Parliament Speaker Ali Akbar Hashemi Rafsanjani and including the Revolutionary Guard minister, Mohsen Rafiqdust.

U.S. officials believe that Assad conveyed his strong desire to see the hostage crisis ended quickly and pressed the Iranians to urge Hezbollah's cooperation.

Monday, after seeing a Hezbollah delegation, Rafsanjani publicly dissociated Iran from the hijackers. He told reporters that if Iran had known the hijacking was planned, it would have acted to prevent the incident. Whether this was true or not, the Iranian clearly felt under pressure to make this statement.

The Beirut hostage episode could prove a turning point in Syrian-Iranian relations if the Amal militia and Syria crack down on Hezbollah and its Iranian-influenced allies in Lebanon. Any lessening of cooperation between those two countries could also help the United States' campaign to curb Shiite terrorism.