In agreement with everything DT said, it's mostly a matter of preference at this stage in the game.
I also agree that straight eyes make tying a bit more pleasant for patterns with portions meant to be tied in under the shank.
Straight eyes, for me, are typically, for streamers, associated with the more 'standard' patterns for salt and warmwater: clousers, deceivers, etc. Most stainless salt hooks are straight eyed.
Down-eyes, again, concerning streamers (since that's what you asked about), I typically associate with classic trout streamer patterns: bucktails like the black nosed dace & mickey finn, the black ghost, etc. Like DT, I also typically tie my buggers on down-eye hooks as well, mostly out of habit.
Up eyes, IIRC, were originally intended to be used when the angler tied his streamer on with a Turle knot, for which the up-eye would allow the standing line to extend out, parallel to and in line with the hook shank, supposedly for the truest action. These days, they're typically more of a nod to tradition, with strong ties to the classic steelhead and salmon streamer patterns.
In terms of other styles of fly, I've used up, straight, and down eyes for dries, nymphs, wets, and streamers, and found little significant difference. That being said, out of habit, I tend to tie standard-dress dries ("Catskill"-based) with down eyes, for a more traditional look, parachutes, hairwings, and foam-bodied dries with straight eyes, and skating flies with up-eyes. Still on the lookout for some cheap K3As to try inverted caddis ties.
Nymphs...I only go through 6 or 8 per year, so I don't tie enough of them to have a preference. That said, I bought a ton of old mustads at an estate sale a year or two ago, so I have hundreds of straight eye nymph/wet hooks, to last me years and years.
Wets, I tie mostly for my own enjoyment, and just fish whatever I have. The only eye I dont use here is straight, with up-eyes lending an interesting look, and down being the classic standard.
Bends are a whole different can of worms...