Next Wednesday morning, one of 12 men will wake up as the freshly minted Democratic nominee for the U.S. Senate in Maryland, only one election away from membership in the country's most exclusive political club.

This fellow will begin the general election campaign with one striking advantage over his Republican opponent: more than two-thirds of the Maryland voters who go to the polls in November will be registered as Democrats.

But, whoever it is, the Democratic nominee also will have one massive problem. He will be among the best kept secrets in state political history. Hundreds of thousands of Maryland voters may all be asking the same question the day after the May 13 primary. They will ask: "Who is this guy?"

The reason is that the Democratic nominee -- be it one of a gaggle of state legislators or a man whose Watergate office was bugged -- most likely will face off in November with Sen. Charles McC. Mathias, a national figure who is expected to easily win the GOP primary again and whose popularity pierces deep into the ranks of the Democratic constituency.

"The thinking is, 'Big deal,' even if you win the primary," said one party official. "Then you gain the dubious distinction of losing to Mac Mathias." o

Still, for several months now, the 12 Democratic candidates have been roaming the state in search of votes and money and public exposure. The money has been hard to come by, so hard that Mathias has raised more than all of them combined. Public exposure at times has been even more difficult. At more than one candidate forum, the contestants seemingly have outnumbered the audience.

It has been a race where everyone is a dark horse, where each man has been groping for that bit of magic that will make him a winner. Who will win? Harry Hughes, a former dark horse himself who is now Maryland's governor, was recently asked that question. Responded Hughes: "I haven't the faintest idea."

Most state political observers believe the winner will emerge from among five candidates: state Sen. Edward T. Conroy of Prince George's; Sen. Victor L. Crawford of Montgomery, Sen. Robert L. Douglass and Del. Dennis C. McCoy of Baltimore, and R. Spencer Oliver of Montgomery, who has never held office but has served the federal government in various capacities and once was national president of Young Democrats.

There are seven others running: Frank J. Broschart, a Suitland teacher; Mello Cottone, a Silver Spring lawyer; John A. Kennedy, a Joppa engineer; David A. Shaw, a Westminster real estate developer; Kurt Summers, a Rockville administrator; Richard Taranto, a Fallston school administrator, and James A. Young, a man of such anonymity that even the League of Women Voters could not track him down.

The voters' lack of interest in all these candidates hasn't diminished their zeal in the slightest, as they virtually trip over one another in the quest for votes. Del. David Shapiro, who was squiring candidate McCoy at a walkathon for Israel in Baltimore last Sunday, suddenly found himself with Olliver at his heels, wanting to be introduced to constituents, and Conroy stepping from a big black car for the same purpose.

Each has been trying to appeal to the voters with his own special brand of politics, and many believe that Conroy, with a statewide constituency stitched together through 18 years of legislative work for veterans, handicapped persons and antiabortionists, may have the edge.

"If he can keep them together, he's got the best chance," said Del. Judith Toth (D-Montgomery) who is supporting Crawford in the race. "The real question is getting them all out to vote."

The 51-year-old Conroy has been moving around the state pitching his strong stand on defense ("We are in severe straits, as far as the defense of the country is concerned. I am deeply concerned.") and seldom letting the audience forget that he was wounded in the Korean war.

In the endorsement derby, Conroy proclaims himself the winner, ticking off the support of various state legislators, the endorsement of several labor unions and the backing of many of the Democratic political clubs that bring out the precinct workers and the vote on election day.

Candidate Crawford, a Silver Spring lawyer with 11 years in the state Senate and his own list of political boosters, is quick to dispute the potency of Conroy's labor endorsements, if not their authenticity.

"Labor support," Crawford said with a laugh recently. "Conroy has no real labor support. Labor wants Mac Mathias too bad for that." The Republican incumbent has received more than $50,000 from labor's political action committees.

Crawford, a 48-year-old liberal who has gone off in search of higher office more times than he cares to remember, has poured $34,000 of his own money into the race, and because of that, has the largest campaign war chest of all the candidates.

"We're going to spend $60,000 on media," he recently asserted with characteristic Crawford bravado. "If you're gonna take a shot in the big leagues, you've got to do it."

His commercials, now blitzing the airwaves in Baltimore and Washington, picture Crawford in variuos legislative poses while an announcer proclaims, "If you want to know where a man's going, look at where he's been."

In his appearances at political forums and living room coffees, Crawford tells audiences: "We're in deep trouble . . . going into a deep recession," and pushes for immediate imposition of wage, price and profit controls.

Crawford says his major opponent is Conroy, but he also acknowledges that the liberal Oliver has cut into his traditional Montgomery County base of support.

Oliver, meanwhile, who suddenly became a major factor in the race with the Baltimore Sun's weekend endorsement, focuses on his background as general counsel and staff director to the congressional commission monitoring the Helsinki Accords -- a far-reaching pact on European security, economic cooperation and human rights.

"He stood up to the Russians," Oliver's campaign literature proclaims in bold, black type, referring to a speech at the 1977 Belgrade Conference, in which he attacked the Soviet Union for human rights violations.

But despite Oliver's foreign affairs and Capitol Hill credentials, voters seem more interested in the fact that while he was a Democratic National Committee official in 1972 his telephone was bugged by the Watergate burglars. In 1976, he received a $215,000 settlement in his lawsuit against the Nixon reelection committee.

Oliver seldom mentions the incident, preferring instead to talk about his stance against the decontrol of oil prices and in favor of establishment of a government corporation to bargain for oil on the world market and thus keep prices down.

Oliver has picked up the endorsements of some of Baltimore's liberal political clubs and several Montgomery County central committee members. And he is campaigning everywhere, even making a foray in his blue blazer and rep tie into a traditional Maryland bull roast at a steelworkers' hall in one of Baltimore's working-class suburbs.

But there, as Del. Louis DePazzo, who represents the district, was quick to point out, "The Sun endorsement don't mean a thing, and people will still say 'What's an Oliver?'"

At that political fund-raiser, while Oliver worked one side of the giant steelworkers' hall, Dennis McCoy, the hometown boy, was working the other.

The 38-year-old McCoy, who is chairman of the Baltimore city delegation in the state legislature and of a special committee on Maryland ports, says these positions have given him a "special understanding of the economic fabric of the state," which would be an asset in the Senate. At forums and other political functions, he emphasizes his liberal stance on urban issues, and recently told a small political club, "We must look at problems in terms of human suffering and human need."

McCoy has the endorsement of Rep. Barbara Mikulski (D-Md.) and Baltimore City Council President Walter Orlinsky, both liberal leaders in the city. His disappointment at losing the Sun endorsement to Oliver was evident when he learned the news, although all he would say was a terse, "I'm not surprised."

The entrance into the race of McCoy's Baltimore colleague, Douglass, destroyed McCoy's hope to come out of the populous city with enough votes to beat his competition. Douglass, who is head of the legislature's black caucus, has raised less than $5,000 for the race and campaigned little, but he is still expected to carry his Baltimore East Side area.

The crowded race, from its start in early April, has generated little enthusiasm and even less attack and counterattack among its participants. But that is changing as the campaign enters its final week.

Since the Sun endorsement, Crawford has begun accusing Oliver of concealing the fact that he ran unsuccessfully in a Democratic primary for the Texas legislature more than 18 years ago.

"Why not tell people this? Why hide it?" Crawford asked at last Saturday's Montgomery fund raiser.

"There are a lot of things a voter ought to know that are important," the enraged Oliver retorted, insisting it was not a secret anyway. "But just because Vic Crawford said I ran for the legislature in 1962, I cannot imagine the significance of that to anybody."

It is hard to tell just what would be significant to voters in a 12-man race among little-knowns and unknowns, many politicians note.

"If anybody tells you they can call this one," said former acting governor Blair Lee III, "tell them they're crazy."