PHOENIX — Battered by the recession and the deepest and most widespread budget deficits in several decades, a large majority of states are slicing into their social safety nets — often crippling preventive efforts that officials say would save money over time.

President Obama’s $787 billion stimulus package is helping to alleviate some of the pain, providing large amounts of money to pay for education and unemployment insurance, bolster food stamp programs and expand tax credits for low earners. But the money will offset only 40 percent of the losses in state revenues, and programs for vulnerable groups have been cut in at least 34 states, according to the Center for Budget and Policy Priorities, a private research group in Washington.

Perhaps nowhere have the cuts been more disruptive than in Arizona, where more than 1,000 frail elderly people are struggling without home-care aides to help with bathing, housekeeping and trips to the doctor. Officials acknowledge that some are apt to become sicker or fall, ending up in nursing homes at a far higher cost.

Ohio and other states face large cutbacks in child welfare investigations, which may mean more injured children and more taken into foster care. Despite tax increases, California has ended dental coverage for adults on Medicaid, all but guaranteeing future medical problems.

“There’s no question that we’re getting short-term savings that will result in greater long-term human and financial costs,” said Linda J. Blessing, interim chief of the Arizona Department of Economic Security, expressing the concerns of officials and community agencies around the country. “There are no good options, just less bad options.”

Arizona has one of the nation’s highest deficits in relation to its budget. As revenues sank late last year, forcing across-the-board cuts this spring, the child protection agency stopped investigating every report of potential abuse or neglect, and sharply reduced counseling of families deemed at risk of violence. Some toddlers with debilities like autism and Down syndrome are not getting therapies that can bring lifelong benefits. And here, as in other states, the drive to help disabled people live at home has been set back.

Mary Beth Thompson, 57, who lives in an apartment with two small dogs here, is on the growing waiting list for help. Seriously overweight, with chronic pain and weakness on her left side, she has trouble moving about and cannot step into the bathtub without falling, she said, displaying the cast on her broken wrist.

“I can’t even walk to do the laundry anymore,” she said from the chair where she spends most of her days playing with her dogs, one of which she has trained to knock the handset off the telephone so she can reach it when she falls.

Winona Conn, 75, who uses a wheelchair because of a paralyzed leg, has been on the waiting list for home aid for a year. “It feels like you’ve been shelved,” she said.

In Florida, recent modest cuts in home aid came on top of a growing backlog, while the number of people in need keeps climbing. State support for home and community services was reduced by $2 million in 2008, and the waiting list has grown to 50,000 from 30,000, said E. Douglas Beach, secretary of the Department of Elder Affairs.

Reluctantly endorsing another $1 million in cuts next year to salvage a different program, Mr. Beach told legislators, “It’s like trying to decide whether to give up your first-born boy or your first-born girl.”

Mary Lynn Kasunic, president of the Area Agency on Aging in Phoenix, described the potential consequences. “If you don’t give people a bath a couple times a week, change the linens and make sure they get their medicines, their health will decline much faster,” she said. “They end up in the emergency room in a crisis, and then in a nursing home.”

The Illinois governor’s budget proposal would scale back home visits to ill-equipped first-time mothers, who are given advice over 18 months that experts say is repaid many times over in reduced child abuse and better school preparation.

“We spend $1.2 billion a year on child welfare,” said Diana M. Rauner, director of the Ounce of Prevention Fund in Chicago, which channels government money to private agencies. “You’d think we’d spend a lot of money to keep people out of that system.”

Ohio’s proposed budget “will dramatically decrease our ability to investigate reports of abuse and neglect,” with some counties losing 75 percent of their investigators, said Joel Potts, director of the Ohio Job and Family Services Directors’ Association, which represents county officials.

New York State is using stimulus money and a tax increase to avoid most of the large cuts in child care, nurse visits to inexperienced mothers and other services that were originally proposed. But if revenues keep falling by the billions, “all bets are off,” said Karen Schimke, president of the Schuyler Center for Analysis and Advocacy in Albany, which studies child and family issues.

As in many states, Arizona’s crunch came on fast and hard. In January, the newly seated Republican governor, Jan Brewer, had to cut $1.6 billion from a $10 billion annual budget — squeezing all the reductions into the final five months of the fiscal year ending June 30.

Arizona expects a $3 billion shortfall in the next fiscal year. In a speech to legislators in March, Ms. Brewer proposed to fill the chasm with $1 billion in spending cuts, $1 billion in federal stimulus money and — in a risky idea she floated after emphasizing her conservative credentials — $1 billion raised through “a temporary tax increase.”

Some Republican legislators still argue that state expenses are too large, while officials say that carving another $2 billion from the budget will wreak havoc. Ms. Blessing, of the Department of Economic Security, said her agency had already laid off 800 workers, including 15 percent of its child protection investigators, and imposed furloughs amounting to a 10 percent pay cut.

In one bit of good news for the department and its clients, the state has secured $18 million from the stimulus package to save child care subsidies for the working poor.

But some efforts to prevent child abuse, like in-home counseling of troubled families, have been deeply cut. This presents investigators with a stark choice: either remove children and put them in foster care or, as one case worker put it, “wait for something bad to happen.”

Idolina Moreno, 36, and her five children are still together and happier, she says, because they have been visited weekly for the last several months by a counselor who defused a simmering crisis. One daughter was angry and violent, Ms. Moreno said, and badly bruised the infant boy; Ms. Moreno admits to throwing a plastic bat to stop her. A school nurse called Child Protective Services.

Instead of removing the children, the agency called in a counselor who meets with family members both individually and together. “She’s been wonderful,” Ms. Moreno said.

Officials said it appeared likely that the counseling will continue for now. But she has also been told that special therapies for her mentally retarded 6-year-old son may be eliminated. “I don’t know what I’ll do if that happens,” she said. “I’m really worried.”