With his independent presidential candidacy now two months old, John B. Anderson is battling money and recognition problems to stay in the political big leagues.

Consider, for a moment, Thursday night. Jimmy Carter, Air Force One and the entire White House press corps were in Rome, where the president met the leaders of the Italian government with great fanfare. Ronald Reagan was in New York, speaking to a crowd of 1,500 persons, each rich enough to pay $200 for a plate of routine banquet food.

And where was Anderson? He was in Sandra Featherman's sparsely furnished living room in Philadelphia, talking to 46 persons.

Everyone there was well-placed and earnest. Although Featherman, a Temple University political science professor, would have liked a bigger crowd, the event raised almost $6,000. A later event raised more. Everyone in the Anderson campaign thought the night a big success.

But in the world of presidential politics, $6,000 is small potatoes. Herein lies a potential major problem for Anderson: Since he announced his independent candidacy, Anderson's campaign has raised $2.5 million, or $310,000 a week, according to fund-raising spokesmen.

This is an impressive feat for a congressman from Rockford Ill., that few people had ever heard of until a couple of months ago. But if the campaign keeps raising money at that rate, it will have raised only $8.8 million by the Nov. 4 election day, or less than one third of the $29.4 million the Republican and Democratic nominees will automatically receive in public money.

The campaign is stepping up its fund-raising efforts in hopes of raising an additional $5 million, or $500,000 a week by Labor Day.

Meanwhile, it is struggling with another problem: How to keep interest in the Anderson candidacy alive over a summer when the major parties hold conventions that will give their nominees millions of dollars worth of free publicity.

Anderson's immediate response has been to schedule a trip to Europe and the Mideast in early July, about the time of the Republican National Convention.

Anderson press aides called major news organizations last week trying to stir up interest in the outing by saying West German Chancellor Helmut Schmidt, Israeli Prime Minister Menachem Begin and Egyptian President Anwar Sadat had agreed to meet with Anderson.

This apparently was premature. Anderson said yesterday that feelers have been put out to these men as well as British Prime Minister Margaret Thatcher and French President Valery Giscard d'Estaing, but arrangements are not set.

Attention is vital to the Anderson campaign and its fund-raising efforts. During the last month, he has consistently drawn the support of 19 to 23 percent in national polls when matched against Carter and Reagan. He has to at least maintain that standard throughout the summer to remain a serious contender.

"Fund-raising is never easy. There's only a small amount of people who ever give political money," said Featherman, who has staged fund-raising efforts for Republican and Democratic candidates in Philadelphia. "When I go to political fund-raising events, I see the same old faces time after time.

"I know there were people who didn't come to my house because they didn't want to put money in a campaign that wouldn't be successful," she added. "There are still a great many doubters out there."

Anderson, of course, realizes this. Every place he goes he cites the latest favorable polls results and pledges "this isn't just some kind of quixotic adventure."

"I really believe this is not one of those lost causes," he said Thursday night. "This isn't going to be a campaign where we become invisible while the other candidates blanket the tube . . . you're not going to be ashamed of us in November."

This week's favorite poll was one by Louis Harris. It reported that when asked if Anderson had a serious chance of winning the presidency, he drew 31 percent of the vote compared to 35 percent for Reagan and 31 percent for Carter. In addition, he outdrew the president in large areas of the industrial northeast.

This week also found the campaign trying to deal good-naturedly with what has been called its "chablis and brie" image -- a reference to the alleged favorite eating and drinking habits of Anderson supporters. The campaign's answer: Serve chablis and brie on the press bus.

According to campaign treasurer Francis Sheehan, all but about 13 percent of the $2.5 million Anderson has raised since becoming an independent candidate on April 24 has been through direct mail appeals.About 85 percent of the people who donated to him as a Republican candidate have now made additional contributions and the campaign has built up a donor list of 135,000.

In each of the next three weeks 500,000 more fund-raising letters will be sent out. In addition, Harvard law professor Laurence Tribe is preparing a legal challenge to the federal election finance laws that, if successful, would qualify Anderson for public money in direct proportion to the amount of vote he receives in November. Anderson once was an ardent supporter of the law he is now trying to overturn.