As experimental tools in the physical and life sciences become increasingly sophisticated and costly, there is a need to optimize the choice of experimental parameters to maximize the informativeness of the data and minimize cost. When designing a scientific experiment, an experimentalist often faces a choice among a suite of data collection modalities or instruments with varying fidelities and costs. Analyzing the tradeoff between high-fidelity, high-cost measurements and low-fidelity, low-cost measurements is often difficult due to complex data collection procedures and budget constraints. Here, we propose an approach for designing such experiments using Bayesian power posteriors, which naturally account for instruments with varying fidelities. Whereas existing approaches for multi-fidelity experimental design are often bespoke for particular data models and involve complicated inference schemes, our approach using power posteriors is generically applicable for any probabilistic model and straightforward to implement. We show that our approach can be combined with a model of experiment cost to allow for multi-fidelity experimental design. We demonstrate our approach through a series of simulated examples and an application to a genomics experiment.