When Mongolia was a communist state, how did the government portray Genghis Khan? Did they ever manipulate his image for propaganda purposes?

The government (and by extension, Moscow), tried to suppress any glorification of Genghis as a threat to the system.

This is from Alan Sanders's "Historical Dictionary of Mongolia":

In the 20th century, the demands of communist ideology came into direct conflict with historical tradition. After the onslaught against Lamaism in the 1930s and the hardships of World War II, at a time of East-West confrontation when Stalinism was at its zenith, the party returned to matters of ideological reliability. In a resolution issued in 1949, the Politburo of the ruling Mongolian People's Revolutionary Party (MPRP) noted in the teaching of history and literature in schools a tendency "to deviate into bourgeois nationalist views." This was allegedly due to the lack of textbooks written from the Marxist-Leninist viewpoint and confusion among intellectuals over the difference between national pride and "bourgeois nationalism"-boasting about the conquests of Genghis Khan and idealizing Mongol "feudalism" to the point of negating the achievements of the people's revolution in Mongolia, that is, the legitimacy of the party itself.

That said, it seems that the Mongolian leadership still viewed him as a national hero, and during the Khrushchev Thaw an attempt was made in 1962, the 800th anniversary of his birth, to erect a statue (apparently the first statue of Genghis Khan ever erected by Mongols) of him in Deluun Boldog, Khentii (where Genghis was alleged to have been born).

Daramyn Tomor-Ochir, a member of the MPRP Politburo and Secretariat, was given the task of organizing the national celebrations, including the unveiling of a monument at Genghis Khan's supposed birthplace at Deluun Boldog, with a speech by a leading historian for the occasion on 31 May and the publication of a set of Genghis Khan anniversary post-age stamps. At about this time, however, Genghis Khan was attacked in the CPSU (Soviet Communist Party) newspaper, Pravda, as a "reactionary" whose "Mongol-Tatar yoke" of tenor and extortion had laid waste to Russia. In a damage-limitation exercise, the MPRP Politburo. headed by First Secretary Yumjaagiin Tsedenbal quickly issued a new and negative evaluation of the founder of the Mongol state and deprived Tomor-Ochir of his party posts. The new official view of Genghis Khan was that, in founding the Mongol state, he had played a positive role in unifying the divided tribes. "But all his further activity was doubly reactionary and aimed at the seizure of foreign lands, the mass annihilation of the peoples of enslaved countries and the destruction of their material and cultural values." Thus to "deny or insufficiently emphasize the reactionary nature of Genghis Khan's activities is in essence to deviate from the party's position of principle and to encourage nationalism."

Of course since the revolution in 1990 Genghis became ubiquitous. The former Sukhbaatar Square (named after one of the main leaders of the 1921 communist revolution) in Ulaanbaatar lost Sukhbaatar's mausoleum in 2005, a monument to Genghis Khan and the Mongol Empire built in its place, and even lost its name and in 2013 was officially renamed Genghis Khan square.