Use Cross tabulation and Chi Square test

Dear Raphael,

You need to be careful with language here. Influence might imply causation for some people, which you cannot show without something like a randomized control trial.

I agree with Abdulrazzak's advice in most circumstances. Chi-sqaured is a test of association for categorical variables , not of causation or influence. However it only works with one DV and one IV at a time, so you would need to manage the risk of Type I errors, maybe by using a Bonferroni correction to adjust your significance levels.

If your independent variables are binary then you might be able to use ordinal logistic regression with all your IVs and one DV. Alternatively, you might be able to run an explanatory factor analysis and a reliability analysis to turn them into one or more scale variables. The same might apply to your DVs if they are measuring similar things.

Mutivariate analysis   are not suitable for ordinal data. The best in your case and because of having the lowest scales (nominal and ordinal scales) is to conduct elaboration of variables. 

Keep in mind that elaboration of variables is a causal analysis in which you test a relation between two variables after you introduce control variable(s) as test factors  using Chi- Square test.

Elaboration of variables (Crosstabs  with control variable(s))  needs sophistication, knowledge and expertise. The results are surprising. With elaboration you get a separate cross-tab for each category of the control variable(s).

A common method when comparing two sets of variables (and as Peter said if you want to talk about causation more issues are involved, see for example Pearl, 2009) is canonical correlation (or set correlation), but in its standard form has many assumptions. The Gifi group has a version for this for categorical data. It is implemented in the homals package in R (and the homals function, in other releases the function canals might be appropriate ... Their functions exist in lots of packages so names can change). There are lots of other multivariate procedures available for categorical (and ordinal) variables, but from your description I think this is what you are after (with Peter's caution)