The nation's airlines, facing losses of up to $10 million a day in the air traffic controllers' strike, were whipsawed yesterday be canceled flights and an abrupt decline in business, as thousands of potential passengers made other plans, fearing the worst.

Most airlines canceled about one-third of their scheduled flights yesterday in line with Federal Aviation Administration orders following the 7 a.m. walkout by the Professional Air Traffic Controllers Organization.

A prolonged strike would be another heavy blow to the industry, which suffered a $225 million loss last year -- only the third overall loss in the 40 years that industry-wide statistics have been kept. Even a brief strike would have a noticeable impact because of the cancellation of travel plans at a time when air travel is normally heavy.

The strike comes at the worst possible time for several of the most vulnerable air carriers, particularly Pan American World Airlines Inc. and Braniff International. Brannif, which had a net loss of $64 million during the first half of 1981, has had to get approval of its major lenders to put off until Feb. 1 principal and interest payments on nearly $600 million in debts.

Pan Am announced a 10 percent cutback in domestic and international air service in mid-July, as it tries to recover from operating losses of nearly $200 million in the first six months of this year.

Yesterday, Pan Am said all but 40 or its 223 domestic flights are operating. Braniff was operating at about 65 percent, along with the rest of the industry, and reported that many flights carried more passengers than normal.

That was not true throughout the industry, however, as most airlines said the strike threat kep would-be customers away.

USAir canceled all 160 of its scheduled flights yesterday morning, resuming service at noon. No more than two-thirds of the remaining 722 flights departed yesterday and many were only half full, a spokesman said. "We don't know how many of the afternoon flights were able to operate," the spokesman said.

"There are thousands of empty seats because of no-shows," said Daniel Henkin, Air Transport Association spokesman. ATA, which is seeking damages and penalties of $50,000 per hour against the air traffic controllers union, estimated the possible losses industry-wide at $10 million daily.

But some of this loss was offset in the short run by a boom in business Saturday and Sunday, as airline customers advanced their plans. USAir, which normally carries 38,000-plus passengers a day, had 2,000 more on Saturday and 8,000 more on Sunday.

American Airlines was able to get about 80 percent of its flights off yesterday, said spokesman Paul Haney. In Chicago, American's usual 140 departures were cut to 100.

TWA, on the other hand, canceled 40 percent to 50 percent of its 664 daily departures, and many flights left with fewer passengers than normal.

The smaller communter airlines face a serious threat from an extended strike because of their generally greater debt and financial vulnerability.