The search for extraterrestrial life is a wondrous task and one built upon lots of theories, speculation, and the hope that there will some day exist the right kinds of tools to discern the presence of life elsewhere in the universe (presupposing, of course, that such life does not already exist in our own Solar System -- where we have the tools to find and detect it, if not the wherewithal to actualize missions to do the detection). Until then, scientists and researchers of various disciplines move forward by working on older theories, experimenting, conceiving new theories and working to determine empirical proof that hypotheses derived from the theories are practicable. It is in the spirit of such hypothesizing and theorizing that such questions as "Do moons of gas giants affect the habitable zone?" apply.
To make a long story short, the answer is "no," the presence of moons around gas giants do not in and of themselves constitute any change in the parameters of the habitable, or "Goldilocks," zone of a star. The habitable zone is dependent upon the luminosity of the central stellar object itself, so the distance from the parent star and the inside and outside edges of the habitable zone are all determined by its luminosity level. Of course, this is an anthropocentrically designed theory, presupposing that the living organisms in the far stellar system live under much the same conditions as does life on Earth.
However, if life were to develop under different chemistries or biological constructs, where extreme conditions might dictate a vastly different life-rendering mechanism, then the habitable zone could conceivably extend to throughout the universe, dependent only on the conditions being right for the formation and continuation of living organisms.
But working from evidence at hand, life habitable zones exist around stars in areas that would be conducive to life as human's currently understand it to exist.
Dear Astronomer tackled the gas giant moon question by first giving a breakdown of the famed Drake Equation, the formula presented by Frank Drake in 1961 to attempt to calculate the number of worlds that should contain intelligent extraterrestrial civilizations in the Milky Way. These estimates range from the most pessimistic -- that is, zero, meaning Earth is all alone and exceptional -- to the very optimistic, some 185 million planets (but each of the factors would have to be at a maximum). Still, that would be a rather large number of extraterrestrial civilizations.
Given what is known about the moons of our own gas giants, like Saturn's Titan, not to mention Jupiter's Europa and Ganymede, satellites of gas giants are capable of maintaining not only water in liquid form (Europa is thought to be a massive ice-encrusted oceanic object -- but atmospheres as well. Although some have atmospheres that would not be suitable for human habitability -- Titan's atmosphere is nitrogen and methane -- some do hold oxygen, such as the aforementioned Europa and Ganymede.
However, for suitable habitability for living organisms as we know them, the gas giants in question (or any planet with a moon holding the right chemical mix) would have to orbit their parent star in the Goldilocks region. Again, though, if the definition of life is expanded to include biological chemistries that are non-Earth-like or anthropocentric, it really would not matter where the worlds were (or even if they were moons of gas giants).
Although living organisms have not been detected outside of the Earth's atmosphere, the possibility does exist that organisms could find purchase on extraterrestrial/extrasolar worlds. Even life with the basic building blocks of life found on our own planet. Given the environmental extremes to which organisms have been found to live on Earth -- organisms living in extreme heat, cold, darkness, etc., known as extremophiles -- it is not outside the realm of possibility that the first example of extraterrestrial life will not be multicellular or have any major degree of intelligence.
Still, there is no reason that moons could not be included in Drake's equation to allow for even more worlds where life -- and the possibility of intelligent life -- might exist. At the same time, there is as yet no reason to believe that Drake's equation might ever rise above zero.
But that is why we search. It is why we question.
At present, no extrasolar moons have been detected, owing to the current limitations of humanity's technology. Still, there have been 760 confirmed extrasolar planets discovered, according to the Extrasolar Planets Encyclopedia. And with so many of the Solar System's planets shepherding at least one moon, it would be reasonable to assume that the many planets that circle other stars should also have at least one in attendant.
And if they have a moon, that moon could maintain conditions conducive for life -- either in or out of the habitable zones of their parent stars.