Around the first of every month, thousands of District residents pick up their Social Security and government benefit checks from convenient mailboxes at their front doors or in the lobbies of their apartment buildings.

But thousands of others--at least 3,500 by post office estimates--must walk or ride to their neighborhood post offices and then wait in line to get their checks because their mailboxes have been broken by thieves and vandals. In many cases, these people have lived without operable mailboxes for years.

"Most of the broken mailboxes are in public housing and low-rent private apartment buildings near the main post office at North Capitol Street and Massachusetts Avenue and in Columbia Heights, Congress Heights, Anacostia, far Northeast and Southwest," said Ronald Minor, a U.S. Postal Service inspector in the external crimes division, which investigates mail theft outside of the postal system.

It is against D.C. postal policy for a mail carrier to put mail in broken receptacles, Minor said, and so going to the post office for their mail has become routine for many.

"Living without a mailbox is just a way of life," said Hattie Traynham, a resident of the Montana Terrace public housing complex at 1625 Montana Ave. NE.

Traynham said her mailbox can now be locked, but it has been broken into often in the past.

"It's a nuisance," she said. "If you don't go out there to get your mail, you might not get it. I try to listen out for the mailman. Around the first of the month, a bunch of kids are usually downstairs and when they see the mailman coming they start yelling, 'Mailman!' Then the mothers go out to meet him at the door."

Although the Postal Service doesn't keep records on the number of broken mailboxes in the city, Minor said that during fiscal 1982 a total of 1,221 federal and city government and personal checks were reported stolen from mailboxes here. He said that during this fiscal year, which ends Sept. 30, more than 1,500 such checks have been reported stolen.

Of these, 492 checks reported to the post office as stolen have been city welfare checks, which city officials said normally range from $190 to $800.

A bill that would make it illegal for a landlord to leave mailboxes unrepaired for an extended length of time is among several dealing with housing conditions that are awaiting City Council action. Other pending bills would make it mandatory for an apartment owner to put a lock on the front door of his building and would allow tenants to repair their mailboxes and do other necessary work themselves, deducting the cost from their rent.

Several landlords have said that because of the frequent vandalism it would be expensive to keep mailboxes in working condition in certain areas.

"I think that both the city government and the private sector are basically opposed to the bill," said City Council member John L. Ray (D-At Large), author of the Mandatory Mail Receptacle Act of 1983. "They're not sure that the legislation is going to bring about the desired results."

"You need mailboxes that are more secure" in areas of the city where there is a high frequency of mail theft, Minor said. "Most boxes in those areas are made of aluminum or weak alloy steel. Most of them are thin enough to be pried open and the locks are weak as well."

Mary Williams of the 1400 block of Newton Street NW said she spends $20 a year to rent a small postal box nearby because if she didn't, most of her mail would be stolen and the rest would be strewn along the sidewalk near her home.

"Near check time, young boys hang around in the lobby to see who gets checks," Williams said. "If there's a yellow envelope or anything that looks like a check, they take it. Most of the boxes here are broken, but some people still receive their mail here. The mail carrier tells them that if they are expecting a check, be down there on check day." The mailboxes in the apartment building have been broken into repeatedly, she said.

"I've put locks on it three times," Williams said of her mailbox. "I went to the hardware store and got a little lock and a key. The third one stayed on there three days, then thieves broke it, and after that I got a post office box."

Williams picks up her mail at a post office at 14th and Irving streets NW, about three blocks from her home. "Sometimes, my old legs and limbs ain't working right," Williams said. "But I hobble to the post office, anyway."