For a new airline, New York Air has a serious, albeit enviable, problem: thousands of potential customers each day can't get through to the airline on the telephone to book reservations.

"We have a problem and one that you could fairly classify as serious," James O'Donnell, NYA senior vice president for marketing, admitted yesterday. "All our lines are constantly busy. . . . But it's a problem of short-term duration and a very happy one, compared with no telephone calls at all."

The airline, which began low-fare service between Washington and New York 10 days ago, has been told by the telephone company that the number of attempted calls that could not get through, counted by computer may have reached as many as 45,000 on one day last week and 25,000 some other days.

"Either their computers are broken, or we're just red-hot," O'Donnell said. Even assuming that each attempt was one of five or 10 tries by the same person, there are still a substantial number of frustrated customers, he said, "and we feel bad about it."

O'Donnell said New York Air is in the process of adding telephone equipment, lines and people as quickly as it can. The reservations system will be doubled by mid-January. "We are hoping customers will be a little patient with us," he said.

In the meantime, he suggested that if customers can, they should stop at the airport ticket counters to purchase tickets. So far, NYA has no city ticket office, although that idea is under consideration. Also, within the next 10 days, he noted, many travel agents will be able to sell NYA tickets directly.

Although some of its flights are sold out, lots of seats are available. During its first week, New York Air flew about half-full, considered a good first week. "We are pleased but not satisfied," O'Donnell said yesterday. "The biggest frustration is knowing how much business couldn't get through on the telephone."

New York Air operates 10 flights in each direction during the week, accommodating a maximum of 2,300 passengers on its DC9 aircraft.

Yesterday, New York Air got a sort of belated Christmas present from Eastern Airlines, its major competitor. Eastern canceled operation of its Air-Shuttle yesterday and sent customers over to New York Air.