The High Pass was first created before the First Age by Oromë the Vala, in order to allow for a crossing of the mountains by the Eldar. Later in the First Age the High Pass was used by the Dwarves, who connected their roads (the Great East Road and the Men-i-Naugrim through Mirkwood) with it.
It's worth noting that there are actually two passes at this location. The lower pass is more prone to being blocked by Orcs, hence most travellers using the higher pass outside of those rare occasions when the Orcs were suppressed.
In the Second Age the High Pass was used by the army of Gil-galad and Elendil when they marched to Mordor in the War of the Last Alliance of Elves and Men. After this war Isildur was slain by Orcs watching the way towards the pass.
During the later Third Age, the Pass became dangerous again because of the Orcs, only with the War of the Dwarves and Orcs, which nearly wiped out all Orcs of the mountains, did it became safe again. Nevertheless by the time of the Quest of Erebor the goblins of Goblin Town had burrowed their way back to it, and Bilbo Baggins and the Dwarves of Thorin Oakenshield were captured.