I am not statistician, and I have a simple question. When is better to use classification and when is better to use multinomial logistic regression?
"Classification" is a broad class of models/methods. Multinomial logistic regression is one model in that class. Are you looking to build a classifier to predict future responses or to explain factors associated with different responses? If the former and you have lots of predictors there are probably better methods out there, like random forest or SVM (different methods handle imbalances differently, some better than others). If you are interested in makings statements about how the independent factors (what is the 938, 11, and 7 in your example?) are associated with the classes then multinomial regression will give you statistics about each of them that have a plain English interpretation. Multinomial regression is robust to imbalance as long as you have sufficient numbers in each response group/class (ie > 10).
Generally speaking, if the relationship between the features and the response is well approximated by a linear model, then an MNL approach tends to work best. On the other hand, if the relationship between the features and the response is complex and non-linear, then methods such as decision trees tend to outperform.
Determining which to use can be done by running a model with different techniques, and estimate their performance on test-error with techniques like k-fold cross-validation.
Keep in mind however that interpretation and visualizations are other aspects to consider. Minimizing test-error may not be the most important aspect of your work on a given problem.