NEW YORK (AP) _ Drexel Burnham Lambert Inc. will eliminate about 300 jobs by year's end as the firm continues its effort to cut costs, a spokesman confirmed today.

The dismissals follow similar moves begun earlier this week by Shearson Lehman Hutton Inc., which began laying off employees in a cutback that could total 800 before 1990 begins.

A Drexel spokesman said that over 100 people already have been let go from the firm's fixed income and equities departments. The remaining cuts are expected to come from Drexel's corporate finance and junk bond areas.

Unidentified sources cited by today's editions of the Wall Street Journal indicated that employees are braced for cuts of up to 800. But Drexel's spokesman said, ''I don't see it getting up that high.

Wall Street firms, particularly retail brokerages catering to individual investors, have been under increasing pressure to trim costs in the face of rising operating expenses.

Most are expected to show a decline in trading profits this year due to the stock market's October plunge and the general pressure on takeover stocks, which had been driving the market higher.

Separately, Bear, Stearns & Co. has begun charging customers a ''handling fee'' on all individual stock and bond trades. The firm is among the last of the major brokerages to do so, and a Bear, Stearns spokesman indicated that the firm has not raised its commission fees in 20 years.

The fee, which amounts to $2.50 per trade, is expected to cover costs and add about $1 million a year in income to the firm's balance sheet.