The balance of power has been so (D) in recent decades that I do not know if the question of balance of power ever came up before Jeffords.
The Senate has changed hands more often than the House has in the past 50 years.
However, a situation where the Senate was so close (i.e. either 50-50 or 51-49 in favor of the party opposite the President) so that one Senator's seat switching parties could tip things one way or the other has happened three times. First, a 50-50 split, but a change in Vice-Presidents tipped the balance of power from the Democrats to the Republicans in 2001. Later that year, Jeffords quit the Republican Party and caucused with the Democrats, which gave the Democrats a 50+1 to 49 majority. Third, when Senator Talent defeated Senator Carnahan in a special election in November 2002, and according to Missouri state law assumed the Senate seat as soon as the election was certified, which again gave the Republicans control with a 50 to 48+2 edge (remember at that time, another independent held Wellstone's old Minnesota seat).