There was a short window in the early years of the Cold War when US spy planes could overfly the Soviet Union with impunity. But by the time the SR-71 came into service, that era was over, having ended in 1960 with the shooting down of a U-2 (an earlier Kelly Johnson Skunkworks creation) over Sverdlovsk. Of course, although the U-2 flew at altitudes comparable to the SR-71, it was subsonic. Arguably, the Blackbird's much greater speed would have made it harder to shoot down, but with spy satellites available, the Americans didn't want to push their luck.
Prof. Sergei Nikitich Khrushchev, who became a US citizen in 1999 and whose father was General Secretary at the time, wrote a detailed inside account† of the U-2 shoot down. By the time of the shoot-down, U-2s had been overflying the Soviet Union since 1956. Prof. Khrushchev wrote, "After the first U-2 flight, Father felt the Americans must be chortling over our impotence."
In addition to the U-2, Soviet air defense bagged a MiG-19, piloted by Lt. Sergei Safronov, who managed to bail out but died of his injuries. The incident occurred on May Day 1960, a big Soviet holiday, and it seems they were too drunk to change the transponder codes from April, causing the radars to misidentify the MiG as hostile.
