A logically fallacious argument, as far as I understand should always be invalid - in every possible world.
But take a kid's argument : This is true, because my father said so. On one hand it seems obviously invalid. Such an attitude is never smart (of course, I do not imply a case in which the father is known to be an expert in something, and therefore is a valid authority, but a kid's childish attitude).
However, there is a possible world in which the father of the kid is omniscient and always telling the truth. It seems a logical possibility. But, if it is a logical possibility, then one cannot argue the argument is _logically_ invalid.
A logically valid argument is one that has the property that, if its premises are true, then its conclusion must also be true. It's a nice question how exactly one wants to spell that out, but if we play along with the talk about "possible worlds", then we can say: A logically valid argument is one that has the property that, if, in any given world, its premises are true, then, in that world, its conclusion must be true. On that understanding, the kid's argument is logically invalid. True, there are some worlds in which everything the kid's father says is true. But that is not enough. For the argument to be valid, this has to be so in all worlds, and it obviously isn't. I think the confusion here may be caused by the phrase "always...invalid, in every possible world". Validity is preservation of truth in every world. But what happens in every world isn't really affected by which world you are in, so talk of what's valid in every world doesn't really make sense. What's valid in one world...