The reputation of firms is largely channeled through their ownership structure. We use this relation to determine reputation spillovers between transnational companies and their participated companies in an ownership network core of 1318 firms. We then apply concepts of network controllability to identify minimum sets of driver nodes (MDS) of 314 firms in this network. The importance of these driver nodes is classified regarding their control contribution, their operating revenue, and their reputation. The latter two are also taken as proxies for the access costs when utilizing firms as driver nodes. Using an enrichment analysis, we find that firms with high reputation maintain the controllability of the network, but rarely become top drivers, whereas firms with medium reputation most likely become top driver nodes. We further show that MDSs with lower access costs can be used to control the reputation dynamics in the whole network.