To add to @statman 's advice, I'll go just a bit deeper. First off...if you are worried that correlation may be invalidating or making your analysis problematic...it's not that simple. In fact depending on your responses and the inherent physical/chemical/biological/sociological relationships that may exist within the system, correlation among responses MIGHT be expected. If it's expected, and you don't see it? Then that might be an issue as well.
So one thing I would do is think about correlation types that might be interesting to explore AND there impact on system understanding and the analysis techniques you'll employ. For example, serial correlation of the responses when examined in experimental execution order is indicative of a lurking factor that may be exerting influence on the responses. Bi-variate correlation may be evident and perhaps even expected if there is some inherent relationship between the responses. A more complicated correlation structure (think in n dimensions...) could be discovered using principal components analysis.
Lastly, ALWAYS make sure you look at correlations of not just the responses...but the residuals. That's where the real gold may lie?
