For those of you unfamiliar, the accepted method of fishing a dry fly is to stand down stream, below rising trout, & cast upstream. The idea here is that you can let your fly float down stream with the current, which will help to eliminate drag on your line. This gives your fly a good, natural drift to it. Also, because trout are almost always facing into the current (facing up stream) when feeding on surface bugs, they won't see you as well because you're down stream & behind them.
Casting your dry up stream is usually the best way to fish a dry fly. It greatly increases your odds of hooking up with the rising trout. Plus, because you're coming up from behind, you can get a lot closer to the fish - & the shorter the distance of your cast & drift, the less chance you have of drag affecting your line. Drag is usually a very bad thing when dry fly fishing.
Now, there will be times when you want or need to take a rising trout & the only way to get your fly to them is by casting down stream. It could be because of some obstruction in the way or some peculiarity with the stream in that location; or maybe getting out of the stream & entering back in below the fish will only put them down.
Whatever the reason, if you find yourself casting your dry fly down stream the biggest problem you're going to face is avoiding drag. Since you can't get as close to the fish as you could if you were below them, judging the distance of your cast is pretty important.
If you can, make your fly land just a few feet or less in front of the fish. You'll have to make sure drag doesn't set in too quickly so maybe try to make an "S" cast by moving your rod from side to side on your final forward cast. You can also "check" your cast by pulling back on the rod slightly just before the end of your final forward cast. Either way, get your fly - somehow - to float over the fish with the least amount of drag possible. Also, keep in mind that if you have too much slack line over the fish, you stand a good chance of missing the strike. You see, this isn't an easy game.
You'll also have to be more patient when the trout strikes your fly as opposed to when casting upstream. If you try to set the hook when you see the flash of white from the trout's open mouth (a natural reaction), you'll miss the fish. Instead, wait another second to set the hook, just when the trout starts back down into the water & closes its mouth.
Fishing a dry fly down stream is a lot harder than upstream. Sometime, though, there's no other choice if you want a certain rising fish. Like all things in fly fishing, there are no hard & fast rules. Just always be prepared to adjust your approach & forget about the dogma.