The Alexandria City Council voted last night to allow a 300-foot cruise ship to make Alexandria its home port for regular weekly cruises to the Caribbean and the Bahamas.

Beginning in April the Nautilus Shipping Lines, a recently formed Liberian company, will operate "The Lady Eleanor" on regular 7-to-8-day cruises from the city.

In approving the coming of "The Lady Eleanor" the council cited the successful half-day stay last November of the West German cruise ship "Caribe," which drew 440 passengers and thousands of spectators to the Potomac River docks in Old Town Alexandria.

The only question raised was whether "The Lady Eleanor" might get stuck in the mud, since the "Caribe" had difficulty getting in and out of the dock at the foot of King Street. The "Caribe" drew 17 feet of water and cleared the mud at best by only three feet. "The Lady Eleanor," which the Nautilus company is in the process of buying, reportedly draws 14 to 18 feet.

In other action, the council approved preliminary negotiations to burn some of the city's trash at the Potomac Electric Power Co. generating plant at the north end of the city, about a mile south of National Airport.

The city's own incinerator, one of Alexandria's worst air polluters, will be closed this July. The city is negotiating with Pepco and state air pollution officials to take burnable trash to Peoco's coal-burning plant, which is just completing a $50 million smokefiltering system designed to remove 99.5 percent of particulates from coal smoke.

City Council member Ellen Pickering objected that Pepco "already is the greatest air polluter in the city" and that the filters may not remove all plastics and "acid vaports" emitted from burning trash. Questions about emissions from burning trash are expected to be explored in the negotiations before the council takes final action.

A $10 million processing center would have to be built to separate burnable trash from metals and glass, which can be sold, and nonusable refuse, which would be trucked to the District of Columbia's sanitary landfill at Lorton.

The council also unanimously rejected a proposed amendment to its dog laws that would have banned dogs from all city parks except in small, designated areas.

The existing laws prohibit dogs from being exercised in school yards and require owners to clean up after dogs except in designated "exercise areas."