This post will most likely cause arguments rather than settle them. I've been watching anime for as long as I can remember, so it has definitely played a role in shaping my views of science fiction and writing. I will list MY top 5 anime series or movies, and what I love about them. Which will in turn reveal what I draw from them for writing. If I leave out something that is definitely influential, it's probably because there wasn't room in this top 5. It doesn't mean I don't like it, it just means there's no room on this list. I would love to do a top 50 or top 100 influences one day if anyone has suggestions
Without further adieu, here's the list
5. HOWL'S MOVING CASTLE
A fairy tale with steampunk elements, 'Howl's Moving Castle' is a wonderful movie. From the mysterious Howl to the naive Sophie, and even the wacky Calficer, it's full of memorab;e characters who all play a vital role in the story. The real world is left behind, and you find yourself right in the middle of the action.
Inuyasha is more fantasy than sci fi (a magical well is really just a stable interdimensional time portal ) but it has strong story elements. It boils down to a love story that spans the eons, intertwining two souls forever. There is swordplay and magic and stuff blowing up and all kinds of weird bad guys. Definitely worth checking out.
3. OUTLAW STAR
I know that Outlaw Star isn't the BEST space anime, but it's good and short and full of action. It has existensial crises and the search for treasure, as well as many other reasons to like it. There is FTL space travel as well as integrated biomechanical beings used to pilot ships. Go check it out.
2. DRAGON BALL Z
The Dragonball universe is full of family, friends, and moral conundrums. The sanctity of life is threatened by villains, and the heroes are forced to weigh the worth of the villains' lives. They usually end up killling the threat rather than deal with it later down the road, but not always. And the themes of redemption and transcendentalism are rampant throughout all three of the dragonball series. Plus they beat the living tar out of each other all the time.
1. GUNDAM WING
You knew it was gonna be something with robots
I like Gundam Wing because of its political turmoil, revolution, and action. Most Gundam series contain these themes, but Wing takes it to a whole new level. Not to mention it has some of the coolest Gundams ever thought up. It's long enough to develop the themes and allow for a couple of complicated love stories, but it's short enough to watch in about a week (maybe two) if you watch a few episodes a day. The ending movie "Endless Waltz" is also recommended.
There's my top 5 anime list. I hope these will help you expand into other series and movies. As I said in the beginning, I hope this list causes arguments rather than settles them. Go watch these wonderful stories, and use them to write your own.