Vice President Al Gore sitting as President of the Senate. Click on picture for larger image

The party of the Vice-President of the United States would be considered the majority party. As President of the Senate, the Vice-President presides over the chamber and may vote to break tie votes. He would participate in the vote to organize the Senate and elect its leaders, favoring the candidates of his party.

There is a historical precedent for your question. In 1881, the Senate began the 47th Congress evenly divided. Vice-President Chester A. Arthur, a Republican, voted to give his party the majority status. However, the Senate went through a very chaotic period as a result of neither side having sufficient members to constitute a quorum. Then President James Garfield was assassinated, dying on September 19. Chester Arthur became President, leaving the office of Vice-President (and therefore President of the Senate) vacant. The mid-term elections in 1882 resulted in more Republicans elected than Democrats, thus affirming the Republicans' majority status by a slim margin.

The Senate had another episode of volatility in the balance of power between the majority and minority during the 83rd Congress. The Senate Majority Leader, Robert Taft, a Republican, died in July, 1953. He had led a Senate with 48 Republicans, 47 Democrats, and 1 Independent. His death gave the governor of Ohio the opportunity to appoint a new Senator, and he chose a Democrat. This meant the Senate would have 47 Republicans, 48 Democrats, and 1 Independent. To prevent the switch in party status and therefore the need for a reorganization of the Senate and changes in committee chairmanships, the Independent (Wayne Morse of Oregon) committed to vote with Republicans on leadership and organizational questions. As a result, the Senate divided at 48-48. This gave the Republican Vice-President, Richard Nixon, the tie-breaking vote over organizational matters. Republicans therefore retained majority status.