We study the influence of the various parameters of scenarios of large-scale structure formation on properties of galaxy clusters, and investigate which cluster properties are most sensitive to these parameters. We present a set of large N-body simulations and derive the intrinsic properties of galaxy clusters in these simulations, which represent a volume of 2563 h-1 Mpc3. The cosmological scenarios studied differ in either the shape of the power spectrum of the initial fluctuations, its normalization, the density parameter Ω0 or the Hubble parameter H0. Between each of the simulations, only one parameter is set differently, so that we can study the influence of that parameter on the cluster properties. The cluster properties that are studied are the mass, line-of-sight velocity dispersion, peculiar velocity, intrinsic shape and orientation with respect to the surroundings of the cluster. The present-day rms mass fluctuation on scales of 8 h-1 Mpc, σ8, which is largely determined by the normalization of the initial power spectrum, has a large impact on the cluster properties. The latter, namely the cluster number density, mass, line-of-sight velocity dispersion and peculiar velocity, are also determined by Ω0, though to a somewhat lesser extent. Other parameters, such as H0, the tilt of the initial fluctuation spectrum and the exact shape of this spectrum, are generally less important. Unlike the other cluster properties studied, the peculiar velocity is found to depend on all parameters of the formation scenario. In a companion paper, the properties of the model clusters are compared with observations to try to discriminate between different cosmological scenarios. Using scaling relations between the average properties of the cluster sample and the parameters of the formation scenario, one may try to interpolate between the scenarios studied here in order to find the parameters of the scenario that is most consistent with the data.