Have either party controlled 60 seats or more? If not what was their highest count?
The most overwhelming majority in the Senate was in the selection of Senators in 1866/7. In the wake of the civil war, the Republican party held 57 seats and the Democrats held 9. These Senators were all selected by State Legistatures.
The largest number of seats held in absolute terms is 76 held by the Democrats to the Republican 16 (and 4 seats going to minor parties), in 1936 during the period of the New Deal.
Have either party controlled 60 seats or more?
Comment: There have only been 50 states since August 21, 1959. I'll answer in terms of 60% of the seats rather than 60 seats, and I'll call this "super-dominance". Another key marker is 2/3 of the Senators, the number of votes needed to override a veto. This is a supermajority.
TL;DR: Super-dominance (a party holding 3/5 of the Senate seats) has happened several times. A party has even held a supermajority quite a few times.
There have only been 50 states since August 21, 1959. I'll answer in terms of 60% of the seats rather than 60 seats. The answer to this question is many times, starting with the second half of the very first Congress, in which pro-administration Senators held 18 of the 26 seats (69%). Parties had not yet formed, but factions most certainly had. The pro-administration faction eventually became the Federalist Party while the anti-administration faction eventually became the Democratic-Republican Party. Neither of those parties exist anymore.
The pro-administration faction continued to hold a majority (but not necessarily super-dominant) of the Senate seats through the second and third Congresses. Parties became a recognized thing in the fourth Congress, with the Federalists holding 20 out of the 30 seats (67%). The Federalists were super-dominant in the fifth and sixth Congresses, but then started falling out of favor. In the eight Congress, it was the Democratic-Republican Party that was super-dominant.
The Democratic-Republican Party continued to be super-dominant through to the 18th Congress. By that time the Federalist Party was essentially gone. The 19th Congress saw the Democratic-Republican Party split into pro- and anti-Jacksonian factions, neither of which was super-dominant. These factions became parties in the 25th Congress, in which the Democratic Party (the pro-Jacksonians) held 35 of the 52 seats (67%). After that, no party was super-dominant until the latter half of the 30th Congress, during which the Democratic Party held 38 of the 60 seats (63%).
The Democratic Party dominated until the onset of the Civil War, sometimes holding a supermajority of the Senate seats. The onset of the Civil War resulted in several vacant Senate seats and saw the birth of the Republican Party. The Republican Party held a supermajority of the non-vacant seats by the end of the 37th Congress. The Republican Party continued to dominate, in a supermajority sense, through the 38th to 43rd Congresses, and were the majority party in the 44th and 45 Congresses. Their majority ended with the 46th Congress.
No party would be super-dominant until the 56th Congress, when the Republican Party held 53 of the 88 non-vacant seats (60.2%). The Republicans remained super-dominant (in some cases holding a supermajority) until the 62nd Congress. Control switched back and forth, with no party super-dominant until the 67th Congress, where the Republicans held 59 of the 96 seats (61.5%). The Republican Party remained the major but not super-dominant party until the 73rd Congress, the start of the Great Depression. By the end of the 73rd Congress, the Democratic Party was super-dominant, holding 60 of the 96 seats. The Democrats held 72 of the 96 seats (75%) at end of the 74th Congress. This Democratic supermajority continued through to the 77th Congress.
Control switch multiple times, with no party being super-dominant, between the 78th and 85 Congresses. By the end of the 86th Congress,the Democrats became super-dominant, holding 66 of the 100 seats (one seat shy of a supermajority). This super-dominance continued through to 90th Congress, including a brief supermajority at the start of the 89th Congress. Democratic dominance, and occasionally super-dominance, continued until the 97th Congress. No party has been super-dominant since the 94th Congress.