Walter F. Mondale, Sen. John Glenn (D-Ohio) and Reubin Askew are the front-runners in fund-raising for the 1984 Democratic presidential campaign, while Sens. Gary Hart (D-Colo.), Alan Cranston (D-Calif.) and Ernest F. Hollings (D-S.C.) finished the first quarter of this year with little cash on hand, federal election reports show.

Hart's light showing was perhaps the greatest surprise of the young campaign season. He wound up the first quarter with $25,684 in the bank and debts of almost $30,000, prompting his campaign to promise "dramatic" new emphasis on fund-raising.

Cranston, Hart's rival for the affections of the liberal wing of the party, ended the quarter with $36,465 in cash and $125,000 in debts, according to his report filed with the Federal Election Commission. But Cranston netted more than $300,000 from a fund-raiser in Los Angeles a week ago that was not included in that report, according to a campaign spokesman.

Former vice president Mondale's showing was by far the strongest, in line with his ranking as front-runner in the public opinion polls. He raised more than $2.4 million from January through March, and had $737,000 in the bank.

Glenn was second to Mondale in both fund-raising and the polls. He raised $1.1 million and had $500,000 cash on hand.

Askew, the former governor of Florida who still is little known in the opinion polls, capitalized on the resources of that wealthy state to raise funds far in excess of his current needs. He raised $809,000 and reported $524,000 cash on hand.

Another southerner, Hollings, whose home state, South Carolina, cannot match Florida's wealth, ranks with Askew at the bottom of the early public opinion polls, but is far behind him in early fund-raising. Hollings raised just under $247,000 and spent almost as much, leaving him with $11,349.

Cranston raised $457,000 and Hart $465,000 during the first three months of the year, according to their FEC reports.

Cranston worked hard and spent heavily to achieve his strong showing in a straw poll at the Massachusetts Democratic convention earlier this month, finishing ahead of Glenn and Hart. That showing came just before his fund-raiser in Los Angeles and, his advisers said, was an important way of demonstrating to potential contributors that Cranston is a viable presidential candidate.

Hart, meanwhile, intends to place new emphasis on fund-raising, his advisers said.

"In the next three quarters of the year, we are going to increase the fund-raising effort dramatically," said Kathryn Bushkin, his press secretary. In addition to beefing up his fund-raising staff, Hart will begin his first direct-mail fund-raising effort--100,000 letters--next month. And he plans fund-raisers in New York, Denver and Los Angeles soon.