My short answer is No, they're not too useful, but let me discuss some details, including positive ones.
Categories, especially derived categories, have been appropriate to describe D-brane charges - and not only charges - beyond the level accessible by homology and K-theory. See e.g.
However, I feel it is correct to say that the string theorists who approached D-branes in this way did so because they first learned lots of category theory - in mathematics courses - and then they tried to apply their knowledge.
I am not sure that a physicist would "naturally" discover the categories - or even formulated them in the very framework how they're usually defined and studied in mathematics. And on the contrary, I guess that the important qualitative as well as quantitative insights about the D-branes - including the complicated situations where category theory has been relevant - could have been obtained without any category theory, too.
But of course, people have different reactions to these issues and these reactions reflect their background. And I - a non-expert in category theory - could very well be missing something important that the category theory experts appreciate while others don't.
Most famously, Joe Polchinski - the very father of the D-branes - reacted wittily to the notion that the D-branes should have been rephrased in terms of category theory. In a talk, he spoke about an analogy with a dog named Ginger. We tell Ginger not to do many things and do others, Ginger. What Ginger hears is "blah blah blah blah Ginger blah blah blah".
In a similar way, Polchinski reprinted "what mathematicians say". It was a complicated paragraph about derived categories and their advanced methodology applied to D-branes. What Joe hears is "blah blah blah blah D-branes blah blah blah blah T-duality blah blah D-branes blah."
Some physicists also try to generalize gauge theory to some "higher gauge theory" using category theory but I don't think that there are any consistent and important theories of this kind. What they're doing is similar to the theories with $p$-forms and extended objects except that they don't do it right.
As always, category theory may offer one a rigorous language to talk about analogies etc. - but I don't think that physicists need anything beyond the common-sense understanding how the method of analogies works. So if you learn category theory - which is pretty tough - I think you should have better reasons than a hope that the theory could be useful for physics.
This post imported from StackExchange Physics at 2014-04-05 03:01 (UCT), posted by SE-user Luboš Motl