Sen. Ernest F. Hollings (D-S.C.) promised every teacher in the country a federally financed $5,000-a-year pay raise as the Democratic presidential hopefuls vied yesterday for honors in challenging President Reagan's claim to the schools issue.

As the president carried his campaign on stricter education standards to Albuquerque, Hollings, former vice president Walter F. Mondale and two other aspirants, Sens. John Glenn (D-Ohio) and Alan Cranston (D-Calif.), questioned Reagan's credentials as a friend of education.

All four criticized Reagan for seeking cuts in federal public school money, with Mondale claiming that, until his recent flurry of speech-making, Reagan had gone 20 years "without a single public statement in support of public education."

Hollings said the "crisis" in education, highlighted in a series of recent reports, was so serious that he was putting aside his opposition to direct federal support of teachers' salaries, as well as his recent unsuccessful crusade to freeze federal spending, and was calling for a virtual doubling of federal aid to education.

He estimated it would take $14 billion a year under his plan to give $5,000-a-year raises to all 2.2 million public school teachers and double that amount to those in center-city or other unusually difficult situations.

That is $3 billion a year more than the price tag Mondale put on his own recently introduced education package. Mondale talked about his proposal yesterday at a news conference releasing a report on education from the Center for National Policy, a Washington think tank.

While Reagan has focused his rhetoric on toughening school standards and has reiterated his opposition to higher federal spending on education, the Democrats are emphasizing the federal role.

That is in part a desire to draw the line with Reagan and, in part, a rivalry among them for the endorsement of the National Education Association, largest of the teachers' unions.

The NEA is taping speeches from the Democratic hopefuls this week for showing at its national convention next month in Philadelphia. Mondale is believed to have the inside track for that endorsement.

The NEA has opposed incentive pay for "master teachers," a proposal Reagan has espoused. Yesterday Cranston and Hollings both cast doubt on the practicality of that approach, while Mondale joined Glenn in saying that he might approve it in some form--but only as a part of a general improvement in teachers' salaries.

Glenn, in an interview Tuesday at The Washington Post, said he believed any increase in funding had to be accompanied by "higher standards," imposed by state, not federal, authorities.

"We can't throw a bag of money at this problem and think that it's suddenly going to get solved," he said. "We should not just pass kids along from grade to grade because their little psyches may be warped if they get separated from their peers. But when they get out of school and still can't read or add two and two, their little psyches are going to get warped, I can guarantee you."

Schools should be held accountable for the skills of their graduates, Glenn said.

Cranston told a group of reporters that Reagan was "very vulnerable" on the education issue, because "he has taken a know-nothing approach."

Mondale spelled out the indictment in his statement, saying that if the president "had had his way, 2 1/2 million American children would have been removed from basic skills classes, half a million handicapped children would have been put out in the cold, a quarter of a million work-study students would have been cut off, 1 million American students couldn't get grants for college, 600,000 graduate students wouldn't get loans, and our National Science Foundation and other federal agencies would pull the plug on thousands of advanced research projects."

Hollings said his proposal would "go a long way toward making teaching a profession...If we want good teachers, we will have to pay for them. It is as simple as that."

His plan would put federal funds more directly into every school budget than ever before, but Hollings insisted it was the only quick way to slow the outflow of talented teachers into better-paid fields. CAPTION: Picture, SEN. ERNEST F. HOLLINGS . . . wants $14 billion more for education.