My impression is that most mathematicians these days who work outside of mathematical logic would agree with the following statement: ``If at all possible, one should try to prove theorems within ZFC, or at least within ZFC plus some mild large cardinal axioms (e.g. the existence of inaccessible cardinals).''
I think this is even true in model theory, and I admit to having this bias myself -- partly because I generally want to prove things using as few hypotheses as possible, and partly just because ZFC is the system that I'm most used to. (I say this as someone who proved a result using Martin's Axiom in my thesis, and then was very happy when I later found a ZFC proof.)
To give an example of this attitude, in the 1970's pure model theory seemed to be getting more ''set theoretic,'' with natural statements such as Chang's Conjecture being proven to be independent of ZFC. I've heard that some model theorists were grateful to Shelah for demonstrating (in his 1978 book Classification Theory) that in fact there still were deep model-theoretic results that could be obtained in ZFC alone. The strategy was to define classes of well-behaved theories (e.g. the superstable theories) where one could prove within ZFC that the category of models is ''nice,'' and then show (again in ZFC) that the models of any theories not satisfying these tameness properties are ''as wild as possible.''