Please be welcome to study our publications dealing with the pertinent multicollinearity issues. Keywords to access the publications here on RG under my research: panda (China), bear (Italy), oak (Spain/Australia).
Are you using the stand-alone MaxEnt software or do you call it from R? In the latter case you can calculate the Pearson's correlation matrix of your environmental raster layers by function layerStats() of package "raster". If you need multicollinearity measures like Variance Inflation Factor or Condition Number, you should extract the raster values (e.g. by as.data.frame() or as.matrix()), and calculate kappa() for CN and usdm::vif() for VIF().
If you want to use your trained MaxEnt model for extrapolation (e.g. climate change impact assessment), then I strongly recommend you to check correlation and multicollinearity before training the model and either drop some of your highly correlated environmental variables or create principal components by PCA.
Also, if you use your model for understanding the ecology of the species by searching for the most important predictors, you should never let the modeling algorithm choose arbitrarily from a corralated predictor pair.