New York — The sign on Mayor Edward I. Koch's desk says, ''Be not afraid.'' Perhaps it is directed at those who come to visit him in City Hall. It's hard to imagine New York's 105th mayor as ever appearing afraid.
Often described as ''feisty,'' ''ebullient,'' and ''the quintessential New Yorker,'' Mayor Koch is prone to take on anything that stands in his way . . . or anyone who has the effrontery to suggest he is not doing a good job managing the nation's largest city.
''I depend on the good common sense of the people . . . to see what is happening and to judge me on the basis of 'Could anybody have done it better?'
''Was the city at the edge of bankruptcy?'' he asks in an interview in his City Hall office. ''Did I keep it out of bankruptcy? Have I provided services to a far greater extent than anybody would have believed possible at this point in our city's life span?''
Mayor Koch says he believes that, given all the available political aspirants , the people of New York would conclude that he is the best one for the job.
''I think I did it better than any of (the others) could. And I intend to keep it up.''
In fact, the mayor's presence here is ubiquitous. His picture is seen on the jacket of his autobiography, ''Mayor,'' which was just released amid a flurry of press attention. He is the subject of conversations in coffee shops on Manhattan's Upper East Side, senior-citizen centers in the Bronx, and on the ferry from Staten Island. Almost any cabdriver, bus passenger, or park-bench sitter will gladly offer an opinion on Koch.
''He's great,'' a man at an Upper West Side party says enthusiastically. His companion frowns.
''I think he's obnoxious,'' she says.
Even those on the fringes of society sometimes respond to Mayor Koch's baton as he leads the sometimes cacophonous symphony that is New York City. Near the United Nations, a homeless man, surrounded by plastic bags full of his belongings, parodies the mayor. The unkempt denizen shakes a can at passers-by and crows, ''How'm I doing?'' This is the stock line Koch has used in campaigns, when walking the streets of the city's five boroughs, and even when facing angry crowds.
But these days the mayor seems less inclined to ask, ''How'm I doing?'' than to say to people, ''Look at what I've done.''
His gossipy autobiography, which has had journalistic and political circles abuzz over who was the most slighted, paints his first six years in office in generally congratulatory terms. His recent State of the City address, a 70 -minute speech, offered some new ideas - such as a local youth public-service corps and the allocation of $100 million to help ease the housing pinch for middle- and lower-income families. But much of it covered past accomplishments.
In his speech, the mayor pointed out that crime decreased nearly 10 percent from 1981 to 1983. He cited increased employment of women and minorities. He touted the reversal of the city's fiscal crisis.
It is true the city is not ''going backward,'' observers say. The mayor can take credit for bringing the city back from the edge of bankruptcy, they say. And there have been management gains during Koch's six years in office, such as the institution of two-man garbage trucks, which is eventually expected to save the city $37 million.
But the growth has not always benefited city residents, some critics say. Resident employment grew between 1977 and 1982, but the ratio of resident employment to total employment dropped.
And although the mayor says he has kept up services to New Yorkers as much as fiscal constraints have allowed, critics say the tough bottom line has hurt some middle- and lower-income families.
New York is seeing ''the destruction of communities as we have known them,'' says Miriam Friedlander, a lower Manhattan city councilor who is not on the mayor's list of favorite people. She charges that his administration is more interested in big development and gentrification. This is just the sort of criticism that brings the mayor out swinging. He loves to rail at ''elitists,'' ''ideologues,'' and the ''radical left.''
''We are intent on keeping our position as the international capital of commerce and communications, culture and finance,'' says Koch.
''And we're doing it - notwithstanding the ideologues, who don't want it done. (They) perceive the city of New York as a city that should only be concerned about providing social welfare services.'' he says. ''I know that we should and must provide those services in a compassionate way. And we do. But you can't do it unless the tax base allows it. . . . I have said that my job, among many other jobs, is to provide a climate for jobs and profits in the private sector.''
He cuts down ''ideologues'' with a scathing rebuke.
''It is not that they are not as good as I am,'' the mayor says. ''It is not that they are beyond redemption. It is not that they are not good Americans. They are. They just don't reflect the common-sense approach.''
But Koch will not always be able to capitalize on his outrageous one-liners and retorts, critics say. There is grumbling that his book has irritated friend and foe alike. Koch denies that any adverse reaction will hurt his management of the city.
Can New York City be managed?
''I am a good chief executive,'' he says. ''And the secret is the following: Never be afraid of having people in your administration who are smarter than you are. I'm never afraid to be with a smarter person because I can do things that they can't do. But they have had expertise in different disciplines that I don't have expertise in.''
Koch's second rule: He was willing to say ''I'm gonna be a one-term mayor, and for that reason I'm going to be a three-term mayor.
''No matter what has to be done, no matter how tough, I will do it. I believe , in fact, that's my strength. The willingness and ability to say 'no' when 'no' is the right word. . . . If you say 'yes' when you can't afford the expenditure, then you may make a friend, but you may destroy the city.''
Mayor Koch has been criticized for lack of long-range planning. His reply is vintage Koch: ''That's baloney.''
He points to the city management report, issued Jan. 30, to illustrate how far he has come in six years. Based on spending decisions he had made earlier, Koch says, most of his judgments turned out to be sound. There are weak spots - potholes that go unfilled, a high school dropout rate of 45 percent - but he is generally pleased.
The mayor also points to the $35 billion, 10-year capital-spending program that has been put into place to repair this aging city's roads, subways, bridges , and water system. And his program to attract and retain business is ''moving ahead really quite well.''
Still, others in city government call for greater creativity from the mayor's administration. One observer points out that the city management reports are good, but they never take problems further than a year or so into the future. And, the observer adds, you can't run a city by simply putting out fires.
A troubling area for the mayor has been his relations with minorities in the city. ''He has conveyed in style, and in substance to a lesser degree, that he doesn't care,'' says one observer.
The mayor acknowledges that the subject comes up frequently, but he maintains his record is ''superb.'' He says employment opportunities have been widened, he has appointed a significant number of blacks and Hispanics to city-government posts, and his administration ''never has and never will shirk its responsibility to the poor.'' He also likes to underline the fact that New York has seen no race riots under his administration.
''I will tell you there is a better climate in the city than in three previous administrations,'' he says.
Koch says he tries very hard to improve his relations with blacks and other minority people. ''I am comforted by the fact that on so many occasions black leaders on a one-on-one basis will say to me, 'You're doing a fine job.' I say, 'Why can't you speak up and say it?' And they will always say, 'Well, I will lose my credibility if I said it.' ''
Nonetheless, many blacks and Hispanics hesitate when asked about Koch. ''We get promises from the mayor, but I don't see much being done,'' says a woman from the South Bronx.