As someone who works on the continuous (bounded) cohomology of Banach algebras: I think the quote is a way of saying "we don't really know". There are certainly questions which start off with extra continuity/boundedness requirements and turn out to be rephraseable in the "purely algebraic" module categories — - L^2$L^2$ cohomology of discrete groups is one, if I remember correctly, Farber and Lueck have written about this.
I'm prepared to believe what's said about algebraic modules over II_1$\mathrm{II}_1$ factors, although I worry/suspect that one has to work with modules over a nastier algebra. Is this the case?
If you want to compute Hochschild cohomology (with coefficients in the algebra, I guess you mean) then this is just hard. It is not easy to find well-defined projective resolutions of Banach objects (if you pass to some dense subalgebra or submodule then more tools are available, this seems to be the approach adopted in much of NCG aà la Connes).
In fact, given a non-injective von Neumann algebra M$M$ (something like a free group factor will do) then there exists an M$M$-bimodule X$X$, which is a Banach space and on which M$M$ acts continuously, and a continuous derivation M --> X$M \to X$ which is not inner. Which sort of answers your question, though probably not in the sense you meant..meant….
If one restricts the module categories then there is a whole theory of Tor and Ext for Banach modules, due to Helemskii -— though it only works on a relatively small class of short exact sequences. However, for von Neumann algebras things are still hard (see work of Christensen, Sinclair, Smith and others).
