This study focuses on the compression area of a cryogenic air separation unit (ASU). The mechanism of exergy consumption in the compressor was revealed. The influence of the compression ratio and of the isentropic efficiency per stage give arguments for proper choice of these decisional parameters. For the purchase cost of the compressor, an exergoeconomic correlation based on the exergetic product represented by the compression ratio and the isentropic efficiency as the Second Law coefficient of performance was used instead of the common thermo-economic one based only on the cost of materials. The impact of the suction temperature on the compressor operating performance is shown, making the gap between the compression stage and the associated intercooler. After optimization of the global system, a specific exergy destruction is assigned to each inter-stage compression cooler. To fit this optimum exergy consumption, a design procedure for the inter-stages and final coolers based on the number of heat transfer units (NTU-ε) method and the number of exergy units destroyed (NEUD) is shown. Graphs are provided that make the application of the method straightforward and much easier to use compared to the usual logarithmic mean temperature difference. A 25% increase in the compression ratio per stage leads to a decrease in the exergy efficiency of 3%, while the purchase cost of the compressor rises by 80%. An increase in the isentropic efficiency of the compressor from 0.7 to 0.85 leads to an increase in the exergetic performance coefficient of 21%, while the compressor purchase cost triples.