The Reagan administration's sweeping new immigration proposals have left more than a few employes at the Immigration and Naturalization Service shaking their heads . They can't handle the workload they have now, much less process the millions of illegal aliens who are likely to file for amnesty.

The problem is a lack of computerization. The INS method of record-keeping "is a non-system designed for the days people came over on steamships," according to former attorney general Griffin B. Bell, under whom the Justice Department tried without success in the last administration to upgrade the system. "There's no hope for INS unless it gets computerized."

Each year INS processes 10 million non-immigrant visitors to the United States, including tourists, businessmen and students. About a million illegal aliens were detained last year. Some 2 million petitions were filed by aliens trying to adjust their status or bring in a relative. Half a million legal immigrants entered the country, and about 170,000 new citizens were naturalized.

Each transaction involved paperwork, and at the INS that's exactly what it is: All the information is collected on paper.

Of the idea of trying to legalize up to 6 million illegals under that record-keeping system, John P. Kratzke, a management expert shepherding INS into the computer age, said: "We might as well just close the service right now."

Doris Meissner, the acting commissioner, agrees. "We're in no position to handle that now," she said.

A rush to computerize during the Carter administration was blocked by Congress for lack of planning. "We were criticized for using computers to mirror our paper files, rather than replacing them," Robert A. Kane, associate commissioner for operations support, said in a recent interview. "Before, we concentrated on controlling the paper files. Now we're looking for ways of controlling the information."

The task is not easy. A master index to keep track of files has 14 million names in it now. Kratzke says that keypunching the index took the equivalent of 680 employes working a full year.

Questions arise about the need for some of the information now required by law. For instance, about 5 million aliens each year send in postcards reporting their addresses. INS was so behind in its work this year that the cards are stacked, unexamined, in a Washington warehouse.

The horror stories from customers are as endless as the lines at the district offices. Phones and letters go unanswered. Files are lost. Routine applications take months or even years to process. "The inconvenience and hardship are an ironic disgrace to a country whose reputation for being able to get a job done undoubtedly attracted many of the aliens here in the first place," a House Government Operations Committee report on INS record-keeping said last fall.

How did INS get in such a fix? In self-defense, its officials note that changes in the immigration laws in the mid-1960s added tremendously to their workload. And although INS has 10,000 employes and a $360 million budget, it has long been the un-wanted stepchild of the Justice Department.

INS has been a constant target of budget cuts as well, perhaps because it has no constituency. Aliens and refugees don't vote.

Several INS officials mention the name Farrell when asked to name a reason for the service's record-keeping problems. Raymond F. Farrell was commissioner of INS from 1962 to 1973, and a friend of Rep. John J. Rooney (D-N.Y.), chairman of the House appropriations subcommittee that controlled the INS budget.

During the period, officials said, Farrell never asked for any money and Rooney never held any real oversight hearings. It was a comfortable, don't-make-waves relationship, and as a result, Carter administration commissioner Leonel J. Castillo said, "we lost a decade. Instead of upgrading existing systems, we were buying our first."

Leonard F. Chapman, a Marine Corps general who succeeded Farrell, began the record automation process in the mid-1970s, and Castillo tried to continue it with the program Congress blocked.

Besides the master index, INS has churned out 2 million computerized, counterfeit-proof "green cards" since 1977 for aliens who become permanent residents. The agency also has permission now from Congress to develop automated systems to control its naturalization applications and its deportation docket.

Another aid, officials agree, would be passing an "efficiency package" that has been pending in Congress. It includes legislation to eliminate the annual alien-address report and the requirement to have two witnesses testify at each naturalization hearing.

The proposals to cut paperwork requirements are threatened, however, by State Department plans to seek legislation waiving visa requirements in several countries, INS officials said. While that proposal would help U.S. embassy consular offices abroad, they say it would just pass on to INS the job of collecting information on the arriving visitor.

Meanwhile, the agency struggles to keep from falling farther behind. When INS makes an alien a permanent resident or citizen, for instance, its workload often multiplies because each new resident or citizen may then apply to bring in a spouse or close relative.

"We can't keep ahead," Meissner said.