Housing starts fell 6 percent in April while personal income rose a weak 0.3 percent, the government reported yesterday, providing strong evidence that the recession continued to deepen last month.

Not only did housing starts fall to a seasonally adjusted annual rate of 881,000 last month--the lowest level since November, the Commerce Department said--but the figure for March, which originally showed a 2 1/2 percent increase, was revised to show a small decline.

Personal income rose to a seasonally adjusted annual rate of $2.531 trillion, up from $2.522 trillion in March, largely on the strength of a $5.4 billion increase in income from interest and a $4.9 billion rise in government transfer payments such as unemployment benefits. Spreading unemployment and a reduction in hours worked more than offset increases in wages and salaries for the month.

Despite the bad news, Treasury Secretary Donald T. Regan maintained that the economy is "starting to inch ahead." But Regan, who was interviewed yesterday on NBC's Today Show, warned that, unless Congress acts quickly to reduce large prospective budget deficits, "our recovery is going to be a very anemic one. We're going to continue with high rates of interest unless we do something about those deficits."

Regan also disagreed with the prediction last week by William Niskanen, a member of the president's Council of Economic Advisers, that unemployment likely would go above 10 percent before declining. "I'm not sure it will go that high," Regan said. "I think that unemployment is peaking about now."

The unemployment rate rose from 9 percent in March to 9.4 percent in April, the highest level since the Great Depression. Regan said he expects the rate to be "in the neighborhood of 9 or just under 9 by the end of the year."

Meanwhile, presidential counselor Edwin Meese III said that the administration will take "other measures" to force interest rates down if congressional approval of a new federal budget doesn't trigger a reduction in interest rates.

Speaking to a small group of reporters over dinner, Meese hinted that the administration might be reconsidering its consistent support for the tight money policy of the Federal Reserve Board, but he did not say exactly what steps the administration would take if rates do not drop.

Earlier in the day, one of President Reagan's top political advisers, who asked to remain unidentified, said that, if there is no "modification" of the economy in the next 45 days, the administration will need to change some aspects of its economic policy. The aide, however, quickly warned that he was not speaking for the president.

Asked what kind of changes might be made, the political aide pointed to the House-passed Lugar bill, which would provide interest-rate subsidies for moderate- and lower-income home buyers, as the type of measure the administration might endorse. An additional proposal for some type of job training program also might be made, he said.

If the news of the March and April declines in housing starts had been available last month, the president's advisers would have been able to convince Reagan to support the Lugar bill rather than oppose it publicly as he did, the aide said.

The housing numbers clearly were not welcome news for the 2,100 builders gathered here for a board meeting of the industry's largest trade group, the National Association of Home Builders.

"I'm now counting 42 months in our company in red ink," Oregon builder Jimmie C. Taylor said. "During a good year, we would build 250 houses. I bought three building permits the first quarter of this year."

The homebuilders were told that the association's political action committee, Build-Pac, and its lobbyists are making an all-out effort to have the Lugar bill passed by the Senate in the next few days. Association President Fred Napolitano said that, despite negative signals from the president's advisers, he is confident that the president would sign housing aid legislation in time to spur the construction of about 300,000 additional houses during this building season.

Even though personal income rose only 0.3 percent last month, consumers increased their level of spending by $10.5 billion to an annual rate of $2.004 trillion, the Commerce Department said. Personal outlays had fallen by $12.3 billion in March.

Disposable personal income--personal income less tax and certain other payments--rose a stronger 0.8 percent largely because of a drop in tax payments. The decline in April was an estimated $14 billion, but will be about only $5 billion in May. This, in effect, will boost tax liabilities and reduce disposable income by about $9 billion in next month's report.

Meanwhile, the Public Employes Department of the AFL-CIO said the economy is weaker now as a result of President Reagan's program than it would have been without it. Using the economic model of Data Resources Inc., the labor group compared the outlook for the economy with Reagan's program to a forecast of how it would have performed if there had been no business tax cut, a smaller income tax cut, smaller spending cuts and easier money.