The president of bankrupt Braniff Airways said yesterday he taped a telephone conversation with his counterpart at American Airlines last year because of concerns about an alleged campaign of dirty tricks being waged by American against Braniff.

The transcript of that conversation is the central piece of evidence in a civil suit brought by the Justice Department last week against American and its president, Robert L. Crandall.

The suit charges that Crandall attempted to monopolize the Dallas/Ft. Worth market by allegedly suggesting during the call that Braniff and American both raise their ticket prices. . The Braniff president, Howard Putnam, who was not implicated in the suit, rejected the suggestion.

The Justice Department suit asks that Crandall be barred from running American for two years. American and Crandall are fighting the charges, which they say are based "on one offhand remark.".

According to the transcript of the Feb. 1, 1982 conversation, Crandall allegedly promised Putnam that, if Braniff raised its fares 20 percent, "I'll raise mine the next morning."

In recalling the conversation in an interview yesterday, Putnam said that Braniff's attorneys "were concerned about what he might talk about, and their fears were justified, as it turned out."

He said the lawyers had advised that "the senior management of this company should be extremely careful in its dealings with American" because of American's alleged campaign against Braniff that Braniff officials say included giving their flights low priority on American's computerized reservations system--one of two used industry-wide and spreading rumors about Braniff's financial problems months before the airline filed for bankruptcy protection last May.

The Justice Department spokesman said yesterday the probe would include information developed during a federal grand jury investigation that was terminated last week without criminal charges.