Eastern Air Lines' pilots began taking a strike vote yesterday, setting the stage for a showdown late next month that could spell the end for the dominant but financially troubled East Coast carrier.

Eastern's bankers have given the airline until Feb. 28 to obtain signed, cost-saving contracts with its three unions that will keep lenders from moving against the airline. Eastern owes $2.5 billion and has been beset with long-running labor and financial troubles while trying to stave off competition from lower-cost carriers.

Two weeks ago, Eastern imposed new work rules including a 20 percent pay cut for flight attendants, represented by the Transport Workers Union, and laid off 1,010 of them. It has also imposed a 20 percent pay cut for its 17,000 noncontract employes.

On Sunday, negotiations with the Eastern unit of the Air Line Pilots Association (ALPA) broke off, and the National Mediation Board imposed a 30-day cooling-off period. When that expires at midnight Feb. 25, the pilots can strike or management can impose work rules.

Eastern also needs a new agreement from mechanics and ramp workers represented by the International Association of Machinists and Aerospace Workers, but its contract with them is valid through 1987.

Although the airline is in a legal position to dictate new wage terms for flight attendants and pilots, it needs signed contracts to satisfy bankers, and Eastern is not negotiating with either the TWU or the ALPA.

Further, Eastern cannot force reopening of its contract with the IAM, which has shown no interest in new discussions.

"There are real questions about survivability if these things can't be worked out," a source close to the negotiations said.

Larry Schulte, chairman of ALPA's Eastern pilot group, told a news conference here that ballots on whether to strike will be mailed to cockpit crews. He stressed that "the pilots do not want a strike," and said, "The question is not . . . whether Eastern pilots will reach into their pockets and donate reasonable wage and other concessions to help management."

Asked if the pilots will strike, given Eastern's financial problems, Schulte said, "Who ever heard of a strike to take a pay cut?"

Asked if he was saying that pilots would assess new work rules and pilot salary schedules, then decide whether to strike, Schulte said, "Yeah."

He did not provide a direct answer when asked repeatedly what the pilots want, although he criticized Eastern's management and seemed to be calling for its resignation. Calls for the resignation of Eastern Chairman Frank Borman have characterized past labor disputes, but Eastern's board has stood by him.

Schulte suggested that a change in management philosophy to one that "knows how to compete in the marketplace" would help solve the problem. For 10 years, Eastern's employes have made many concesssions to management, only to have management ask for more.

Members of management, Schulte said, "blame deregulation, which gave rise to cut-rate competition. Yet, while deregulation was raining on Eastern, it was also raining on Delta, Piedmont, USAir and every other established carrier."

Those three are enjoying enormous financial success.