Sufficient dimension reduction (SDR) models are very popular in the past two decades in regression analysis. However, most current methods do not have robustness in mind when they were built. Therefore, these methods may underperform when the distribution of the predictors is heavy-tailed or when outliers exist in the data. We introduce robust dimension reduction methods using the concept of influence function (IF). An influence function measures the sensitivity of a functional to small perturbations from the data. We investigated the IF of several functionals and settled on the one for canonical correlation. Using this tool, we develop a robust algorithm for fitting the SDR model to contaminated data. In addition, we propose to use the IF for selection of the best candidate among various central matrix based SDR estimators. Lastly, an estimator based on IF for the structural dimension of the SDR model is also proposed. An extensive simulation study was conducted and it shows that our proposed robust algorithm works very well for contaminated data, and our IF-based selector chooses one of the best performing central matrices in the vast majority of the examples.