I don't know of such a table. One serious problem in making such a table one is choosing generating sets for the mapping class groups of the surfaces $S_{g,1}$. There is no canonical choice and I suspect no good choice.
Following Lickorish, one can cook up an algorithm that turns any monodromy into a product of Dehn twists, then into a product of twists in the Lickorish generating set, and then a product of twists the Humphries generating set. But remember that the monodromy only matters up to conjugacy -- thus finding the "best" representative, if such a thing, is somehow related to the conjugacy problem. It all sounds a bit dicey.
In practice, I would go to KnotInfo to find out if a knot in the tables was fibered. If there was a particular monodromy I wanted to compute I would look at Stallings theorem, compute the action of the monodromy on the fiber subgroup (which is a free group), and then transform that into a product of Dehn twists. If that didn't work, I would consult the experts, probably starting with Nathan Dunfield or Dylan Thurston. See also their paper "A random tunnel number one 3–manifold does not fiber over the circle" and its many references.
