Transportation Secretary Drew Lewis' mission when he becomes chief executive of Warner Amex Cable Communications Inc. will be to ease the growing pains of two giants moving into the uncertain, but potentially lucrative business of providing cable television service to the nation's cities.

The partnership of Warner Communications Inc. and American Express Co. has proven that it can navigate through the politically delicate cable franchising fray, having been granted more cable urban awards than any other cable operator.

What it hasn't proven and will demonstrate one way or another under Lewis' aegis is how good a business cable television is likely to become. "Franchising is basically over," said John Reidy, an analyst with Drexel Burnham Lambert Inc. "Now is the time to get on with constructing these systems as rapidly and efficiently as possible without straining the willingness of the two partners to lend."

Lewis will take over Feb. 1, succeeding cable pioneer Gustave M. Hauser, who last month said he was leaving to pursue other business interests. On the top of Lewis' agenda at Warner Amex, the nation's sixth-largest cable system operator, will be completing the construction of cable systems in Pittsburgh, Milwaukee, Dallas, and Houston. The firm has 1.16 million subscribers in cities such as Columbus--where the company introduced its highly publicized two-way Qube system--and Cincinnati. Pieces of franchises in Chicago and St. Louis metropolitan areas also have been awarded to the company, as have sections of the boroughs of New York City.

But ironically, the company's franchising success will present Lewis with his greatest challenge. Cable construction is increasingly expensive and time-consuming, and the company has not turned a profit since the partnership was formed in December 1979.

At that time, American Express bought into the then Warner Cable Co. for $175 million. Both companies added $100 million in investments last year and recently raised a bank credit line from $500 million to $800 million after borrowing close to the previous limit.

Reidy said that losses are expected to hit $30 million at Warner Amex this year alone, about double the 1981 figure, and the partnership is likely to have invested more than $1 billion before making its first yearly profit, which may not be until 1984.

"It could turn a profit a lot quicker than most people think," James D. Robinson III, chairman of American Express, said today. But he noted that reaching profitability "has taken longer than our original outlook."

David Horowitz, a high-ranking staff member at Warner, acknowledged in an inteview today that the cable company "has had cost overruns in constructing some of our systems," but said they are within "acceptable tolerances."

Officials of both companies think the profits later in the decade from cable and pay television, as well as from additional services such as home security and business-data transmission, will make the hard times of the early 1980s worth the wait.

But the Lewis appointment also signals that the cable business increasingly views itself as something more than the cottage, technology-driven industry it has been through much of the last decade. "We decided we were better off looking beyond the industry itself," Robinson said. "We wanted something with additional breadth and capacity."

The announcement that the two companies are bringing in someone of Lewis' stature also is considered in communications industry circles as a vital first piece of good news as Warner struggles to revitalize an image that has been tarnished seriously in recent weeks.

Only a month ago, the company's former assistant treasurer, Solomon Weiss, was convicted of participating in a fraudulent stock deal in connection with Warner's purchase of stock in Westchester Premier Theater. In that case, the government's prosecutor said the "real culprit" in the matter was Warner's chairman, Steven J. Ross. Ross, who has not been charged in the case, vigorously denied the charges.

Earlier this month, the company shocked Wall Street when it announced that, because of a slowdown in sales of Atari, its video-game and computer subsidiary, its earnings for the fourth quarter would be half those of a year ago, news that has crushed the per share value of the company's stock by about $25.

On top of that, the company disclosed Thursday that up to eight Atari executives sold chunks of their Warner stock just before the surprising earnings announcement. And, the next day, the company announced that it was closing efforts to purchase Madison Fund Inc. for about $380 million.

On the other hand, American Express is one of the darlings of the investment community and was just recently added to the list of 30 stocks that make up the Dow Jones Industrial average, the first service firm of its kind to garner a niche on that elite securities compilation.

Both remain extremely profitable companies, as Lewis ackowledged in Washington today. Responding to a question, Lewis said Warner Amex "does not have financial problems" and called it "a very well financed company" with a "very bright future."

Lewis also made clear that he understands that long-term nature of the cable investment in his remarks today. A venture is "not profitable when you build a cement plant or any other plant" and only becomes so after it "starts producing a product and you sell it for more than it costs to produce it," he said.

Also under Lewis' supervision will be Warner Amex Satellite Entertainment Co.

The entertainment affiliate operates Music Television (MTV), the critically successful 'round-the-clock popular music channel, and Nickelodeon, a children's programming network. In addition, it founded and is a partner in The Movie Channel, a pay television venture that recently added MCA and Paramount as partners.