American Telephone & Telegraph Co., already the world's biggest business in terms of private stockholder investment, yesterday announced plans to raise about $720 million of new capital in the largest public sale of common stock in history.

In a move designed primarily to improve the giant utility's financial condition - by reducing the amount of investment in AT&T through debt securities and increasing the proportion of equity equipment - AT&T directors said 12 million news shares of common would be sold on or about Nov. 9, with proceeds designed to retire some outstanding debentures.

Although AT&T previously sold two batches of 12 million common shares - in October 1975 and June 1976 - the company's stock price since has increased. The 1976 sale, at just under $55 a share, raised $658 million for the firm while the 1975 sale raised $552 million at $46 a share.

AT&T's hopes of raising $720 million next month are based on an expected market price for the company's tock of about $60 a share, until yesterday the 1977 low.

However, in active trading on the New York Stock Exchange, AT&T common yesterday dived $1.75 a share and closed at a new low of $59. The Big Board volume of 321,600 shares made Ma Bell the second most active issue of the day.

Wall Street analysts attributed the decline yesterday to a reduction in earnings per share that will result from 12 million additional shares of AT&T common being owned by the public. In 1976, the average number of AT&T common shres outstanding was 595.2 million, up 27 million from 1975.

An AT&T spokesman said the company anticipates that the new offering will reduce per-share profits by 6 cents on an annual basis, a development analysts said will continue pressure on the telephone firm's management to boost profitability to erase potential dilution of earnings.

For a year now, AT&T has been reporting quarterly profits exceeding $1 billion, an unprecedented record for private enterprise. For the nine months ended Sept. 30, AT&T reported yesterday, net income applicable to common stock totaled $3.3 billion ($5.30 a share) compared with $2.7 billion ($4.54) in the 1976 period. Revenues rose to $27 billion from $24 billion.

AT&T plans to use proceeds from the stock sale to redeem, on or about Dec. 12, about one-half of the firm's 30-year, 8.75 per cent debentures scheduled to mature on May 15, 2000. The current redemption price of the debentures is 106.30 per cent of the principal amount and will apply to debenture holders of record on Oct. 28.

A registration statement filed yesterday with the Securities and Exchange Commission said a nationwide group of underwriters would sell the proposed offering. The investment syndicate will be managed by Morgan, Stanley & Co.; Merrill Lynch, Pierce, Fenner & Smith; and Salomon Brothers.

AT&T spokesman Jack Gertz said the largest equity sale in history is aimed at permitting his firm greater financial flexibility by reducing the ratio of debt investment. Management of the utility has been aiming to reduce the debt ratio to 45 per cent from an unsually high level in recent years - it was 49.6 per cent at the start of 1976.

The company added significantly to its outstanding debt securities in the early 1970s as interest rates were soaring - adding costs that helped fuel telephone rate increase. A high level of debt ratio also is avoided because it could lead rating companies to "downgrade" the investment qualities of such securities.