A 1994 CBS/New York Times survey asked about the racial composition of poor people, the racial composition of people on welfare, and a number of questions about attitudes towards welfare and welfare policy. I'll talk about the relationship between views of the racial composition of people on welfare and welfare policy--the relationship between views of the racial composition of the poor and policy are similar.
First, the opinions of blacks and non-blacks (I'll say "whites" for short) about the racial composition of the poor are about the same, but a there is a difference in beliefs about the racial composition of people on welfare: 44% of whites say most people on welfare are black, only 20% say most people on welfare are white; among blacks, it's 29% and 27%.
Among whites, people who think that more people on welfare are black differ on many questions--I have put the option they are more likely to favor in bold:
"Would you be willing or unwilling to pay more in taxes in order to provide job training and public service jobs for people on welfare so that they can get off welfare?"
"what is more to blame if people are poor--lack of effort on their own part, or circumstances beyond their control?"
"what is more to blame if people are on welfare--lack of effort on their own part, or circumstances beyond their control?
"do you think that most people who receive money from welfare could get along without it if they tried, or do you think that most of them really need this help?"
"Do you think that most welfare recipients really want to work, or not?"
"Do you think that there are jobs available for most welfare recipients who really want to work, or not?"
"Do you favor or oppose limiting how long mothers with young children can receive welfare benefits?"
"As part of a welfare reform program, do you think the government should create work programs for people on welfare and require them to participate in the programs, or not?"
The answers of whites who thought that a larger proportion of people on welfare were black were uniformly more negative or less sympathetic, with the possible exception of the last question. In most cases, the differences were pretty large: for example, 50% of those who said that most people on welfare said that lack of effort was to blame if people were poor, but only 34% of those who said that most people on welfare were white blamed lack of effort.
However, there were also some questions on which there was no statistically significant difference (and in most cases, not even a pattern suggesting a difference):
"Do you agree or disagree: it is the responsibility of the government to take care of people who can't take care of themselves."
"Do you think that government spending on programs for poor children should be increased, decreased, or kept about the same?"
"Do you think that government spending on welfare should be increased, decreased, or kept about the same?"
"Do you think that most of these jobs [that welfare recipients could get] pay enough to support a family?"
"Do you think that women with young children who receive welfare should be required to work or should they stay at home and take care of their young children?"
"Which of these statements comes closer to your view about welfare reform: welfare recipients in a work program should be allowed to receive benefits as long as they are willing to work for them OR after a year or two, welfare recipients should not be eligible for a work program and should stop receiving benefits?"
"Do you think that unmarried mothers who are under the age of 21 and have no way of supporting their children should or should not be able to receive welfare?"
There seems to be a pattern. Belief that more welfare recipients are black goes along (or went along--this was 20 years ago) with a belief that welfare recipients could get work, but don't want to. However, the result isn't a loss of support for welfare spending, but an increase in support for work requirements (for people without small children).
[Data from the Roper Center for Public Opinion Research]