At its just-concluded meeting here, the Democratic Party did nothing to address one of its fundamental and most troubling problems - its inability to attract the votes of white men.

The party's national committee elected a black man, Ronald H. Brown, as its chairman. And to make its various constituencies happy, the party picked four vice chairs. One was a Hispanic man and three were women - one white, one black, one Hispanic."We're witnessing the demise of the white, middle-aged male in the Democratic Party," George Bruno, former party chairman in New Hampshire, said wistfully. "It's ridiculous."

That may be something of an overstatement. The party positions in question, aside from the chairmanship itself, are relatively insignificant; they carry little power or visibility.

"There's still a role for white males in this party," said Sen. John Breaux of Louisiana, noting the abundance of such Democrats on Capitol Hill and in the nation's statehouses. "I'm still here. My concern is to continue to impress upon people the seriousness of the (white male) problem."

But the makeup of that list of party officials, and the balancing act that produced it, symbolize the dilemma of the Democrats.

The party must fight a continuing battle within its own ranks, even as it ponders how to broaden its base. It must strive to empower the minorities essential to its well-being without seeming to be the captive of those very groups.

"Can a party with conflicting loyalties offer twin or even multiple messages?" asked Natalie Davis, a political scientist and loyal Alabama Democrat, in a paper urging her party to turn 1989 into The Year of the White Male. "If the problem is posed in such terms, then the future is bleak."

And Democratic leaders know that the Republicans will try to exploit those internal tensions, wooing whatever group in the Democratic rainbow has reason to feel under-appreciated.

In some ways, the Democrats are looking increasingly like a party under siege.

They look in one direction and see a new Republican National Chairman, Lee Atwater, a ferocious and proven political professional who has devoted his adult life to bloodying and defeating Democrats. Atwater has been talking about making a real effort to lure members of the traditionally Democratic black middle class into the GOP fold.

They look in another direction and see a revived Republican Congressional Campaign Committee, headed by former White House political director Edward Rollins. That organization is committed to spending millions of dollars on the process of whittling away at the Democrats' seemingly permanent majority in the House of Representatives.

But such downbeat thoughts were not voiced openly. "We cannot, must not, lose our confidence," Brown told the gathering.