The research area known as Reverse Mathematics is concerned with finding out the weakest theory that suffices to prove a given mathematical statement over a very weak base theory. The project has now been successfully carried out for a huge proportion of the theorems of classical mathematics, many of which would seem to be central for any robust effort in applied mathematics. So it seems to me that the answer to your question is provided by the precise reverse mathematical strength of the principal classical theorems used in whatever branch of applied mathematics you have in mind, which I expect might include much of classical analysis and other areas.
There is a particularly good book on reverse mathematics by Stephen Simpson, and the topic has been mentioned several times here on MathOverflowmentioned several times here on MathOverflow.
One surprising outcome of the work is that numerous classical theorems have turned out to be equivalent to each other, grouped in a comparatively small number of equivalence classes. Follow the link above for information about the five principal theories.
