NEW YORK -- Sal DiGiovanni often fantasizes about shaking people's hands. Fat hands, small hands, it never really matters. He grabs them all, and in his dreams, people are always eager to clasp his in return.

It is only a vision, though, and probably will never be more than that, because DiGiovanni is a "sanman," a New York City garbage collector. Few sets of hands are less popular to shake in this city.

"You should see the way people look at me in the morning," said DiGiovanni, who rides a truck on the West Side of Manhattan, picking up 12 tons of trash daily. "You would think I was the garbage instead of the guy who takes it away. I get thanked about twice a year."

So much for the gratitude bestowed on the faithful public servant.

Few residents or visitors would argue with the assertion that New York is a dirty city. It generates more waste by far than any place in the United States. The $650 million spent by the city each year on the Sanitation Department is nearly twice the entire municipal budget of Hartford, Conn.

But the Sanitation Department is in many ways the city's most visible agency. With 12,000 employees, it picks up more than 16,000 tons of refuse each day. Private haulers collect another 11,000 tons. If a police car fails to make one of its four routine swings down a street some afternoon or evening, nobody is likely to notice. Remove a single garbage truck from a single street on any day, however, and the howling commences immediately.

"If we are supposed to be there by 10, they are on the phone at 10:02," said Carl Fera, a supervisor at the massive 56th Street depot on the far edge of the West Side. "The people who think they have a right to this sort of service are in for a new way of life because the money and the men are just not going to be there from now on. And we can't sweep streets or empty baskets unless we have someone to do it."

The administration of Mayor David N. Dinkins (D), struggling under the weight of a budget deficit that could exceed $2.5 billion in the next year, has had to cut services more deeply than at any time since the city's fiscal crisis in 1975. Every city agency has been forced to pare spending, some by as much as 17 percent, and there are reports that the cutbacks may grow larger along with budget shortfalls.

Hospitals so overcrowded that patients compete to enter emergency rooms will be severely affected as will the nation's largest public school system, which even now has trouble finding money to replace tens of thousands of windows smashed by vandals each year. Park maintenance could be decimated and support for the arts is jeopardized.

But perhaps the most immediate and obvious effects of the reductions will be manifest on city streets, where the banner of decay is easiest to detect. The Sanitation Department budget is to be reduced by $49 million this year and by at least $80 million in 1992. City streets now merely dirty could be overflowing with refuse when the warm weather arrives this spring.

"What we are about to see is horrendous," said Raul Juevarez, district associate for Harlem's Community Board 9, which suffers some of the city's most distressing social conditions. "There won't be any street cleaning left. That means in this area, vacant lots strewn with refuse. That means rats and that means disease. I can't paint the picture any darker than it is."

According to New York's own rating system, just half of Manhattan streets are expected to be considered clean this year. In poor neighborhoods such as Harlem and the Lower East Side, the situation will be far worse.

Wire-mesh trash bins rest on nearly every populated street corner in the city. By April, according to the city sanitation officials charged with making sure that they are empty, they will begin to look like refuse-laden ice cream cones, overflowing with garbage. They simply will not be emptied as often as in the past.

Abandoned cars will sit for days before being removed, just as many already do. Refrigerators, couches, tires, anything that can be discarded will rest longer on the curb if it does not fit in the can.

"You could not believe what the citizens of New York expect us to haul away every day," said Andrew Pecoriano, who has spent the last six years working a street-sweeping machine.

"Every type of body part from every animal," he continued. "Clothes, food, cars, appliances. Lots of times they actually think they can just open their windows and toss the stuff into the street. It's like the street is a garbage dump and we are supposed to be there when the stuff hits."

Many people who express concern about the looming garbage crisis remember the fiscal calamity of the 1970s, when the number of street cleaners fell from 2,500 to fewer than 800. Uncontrollable filth became the most obvious symbol of a city losing control. The mess helped to send thousands to live in the suburbs, never to return, and made the nation wonder whether the city could manage civic responsibilities to its residents.

"I don't want to see that come back,," said John Doherty, a 31-year veteran of the department and deputy commissioner for operations. "It was just too terrible. But it isn't going to be easy. My greatest fear is that we will see whole areas of the city where streets are just going to be blocked off by garbage."

He added that the number of street cleaners will dip this year to 700, less than one-third of the 1984 total and fewer than at any time since 1976.

The numbers tell only part of the story, however. A new recycling program, modern collection equipment and a better distribution system have emerged from a department far better managed. Production gains are essential, and in the view of most, including Sanitation Commissioner Steven M. Polan, there is plenty of fat to trim.

But no amount of management can pick up garbage or sweep streets where dirt is constantly replenished. It is the look of the streets that most immediately affects morale of the people in the department.

"I'm taking nothing away for the cops or the firefighters," Fera said. "They are the greatest thing this city has. But we are there every day.

"When crime surges, do they spit on the cops? No way. Well, when the garbage piles up, believe me, you don't hear a lot of sympathy for the sanmen. Unless the city finds some money, it could be a long couple of years around here."