Wood matters!
(Just not as much as pickup placement, pickup type, strings, or hands matter.)
Here's my view on that: ANY component that changes the character of an instrument is important, and any Luthier worth his salt should understand all of the components and how they will ultimately produce an instrument's character. He should definitely understand the basic overall characters of the wood used for an instrument, and then understand how pickup and electronic selection, etc. will enhance, cover up, or change that character.
I'm going to equate building a bass with making wine as a way to exemplify that:
If I was a winemaker, I could use any grape to make a wine, the end product being...well, wine. The thing is though, every grape varietal has a different character. Its this character that the winemaker tries to enhance or bring out in the winemaking process that an experienced wine drinker would appreciate. A good winemaker doesn't make wine for the common guy who just wants to get drunk, he is making a wine for someone who can appreciate its subtleties and nuances. To a casual and inexperienced drinker, these differences aren't apparent. It isn't because they aren't there, its because the drinker just can't appreciate it. You could (erroneously) use their (in)experience of the wine to say "see, the grape varietal doesn't matter, people can't tell the difference. I can use any variety and produce wine, the differences are so small its really only a matter of the additives and alcohol that determines a wine's flavor." To an experienced wine drinker though, the type of grape used is immediately blaringly obvious and important. The argument that the difference between the grape types is small is not the issue, because its precisely that small difference that IS important to one who knows what they are looking for. Therefore, its not really a small difference anymore is it? Do other aspects of the winemaking process effect the end result? Well, duh, of course they do, and its precisely the expert wine maker's job to know what those are, and then to produce the best wine result they can with the grape they start with.
If you are Fender, you don't really care about the wood choice for a bass. You aren't selling to connoisseurs, you're mass producing. If you are a custom builder selling to connoisseurs, you better know and appreciate that EVERY choice you make in a build will add up to the end product...it'll go to your reputation as a builder. A Luthiers' attitude shouldn't be "wood choice is too small of an issue to matter so who cares?", its should be "how can I make that small difference matter even more...how can I work with it and enhance it?"