Airlines seeking customers in the next couple of months -- which are traditionally slow for them -- have reintroduced family discount plans.

Between now and the spring, American, Braniff, United and Trans World airlines -- probably others -- will be offering discounts of 50 percent and 60 percent for immediate family members who accompany a full-fare-paying passenger.

In addition, TWA asked the Civil Aeronautics Board yesterday for permission to offer a "Kids Fly Free" fare that allows a child between the ages of 2 and 17 to fly free on TWA's domestic system when accompanied by an adult flying on a coach fare. Children under 2 already fly free.

Generally under the family plan being offered by the airlines, when one family member pays the full coach or night coach fare, the first accompanying family member pays 50 percent of the regular day coach fare and each additional family member receives a 60 percent discount on the regular day coach fare.

Uniformly, travel on the family plans must begin before midnight March 27 and be completed by midnight May 31. Reservations must be made, and tickets purchased, just 24 hours in advance of the first flight. The lack of a longer advance-purchase period is designed to encourage business or other travelers to decide to take a spouse or other family member along almost at the last minute. There are no minimum-stay requirements.

Donald M. Casey, TWA's senior vice president for marketing, said a family of four combining the family and "Kids Fly Free" fare could travel together for less than the price of two. "The availability of these two fares during the winter months offers families a unique opportunity to take an unexpected winter vacation at a price that is very attractive," he said.

The airlines generally are applying the family discounts to all their domestic routes, although there are some exceptions. TWA, for instance, won't apply the "Kid's" fare on its services to Florida or Arizona, and United won't apply the family fare on services to Hawaii.

Although passenger traffic in 1979 was well above industry expectations, the growth rate fell in December despite relatively strong advance bookings for the holiday period. Domestic traffic grew about 1.5 percent in December, down sharply from November's 7.6 percent increase.

Traffic forecasts for 1980 generally are pessimistic because the airlines and the analysts believe that the economic downturn and higher coach fares caused by skyrocketing fuel costs are expected to keep customers away.