Growing up, none of my friends were into anime. They'd laugh at my obsession with it and most ofthem would make fun of me (some friends right?), but I had a few that actually enjoyed it after I finally managed to introduce them to it.
Since then I've managed to introduce a lot more people to anime and have had varying degrees of success along the way. I've learned a lot and have come up with a pretty straightforward means of introducing non-otaku to anime and increasing my chances of success.
The first step is to think about the types of movies you guys usually go to see. The first person I introduced to anime had a similar taste in movies to me. Which meant I was fairly confident that if I enjoyed a show, she would enjoy it too.
If you find yourselves normally going to see chick flicks, then maybe a slice of life anime would fit your friend better. If you're normally into Super heroes and big flashy battles, try something Naruto or Bleach. If they enjoy supernatural movies and television then maybe try something like Blue Exorcist or possibly start on the "spookier" side of anime.
Next, don't even bother trying to introduce them to a show that isn't dubbed. Chances are that they will flip out and refuse to go on if they're suddenly forced to start reading subtitles. That's something you can ease them into in the future. Trust me, your chances of success go up exponentially this way.
Third, don't fangirl/boy around them too much and don't nag. If they've shown some interest but haven't taken the final plunge, the easiest way to chase them off is to nag them about it.
My housemate and I both enjoy anime, though I'm the bigger fan. A mutual friend of ours discovered Attack on Titan and began going on and on and on about it, constantly nagging us to watch it NOW. I had planned on watching it anyway, so I went ahead and got it out of the way. I enjoyed it, it wasn't my favorite anime ever, but it wasn't too bad.
My housemate, however, absolutely refused. He hates it whenever anyone tries to force him into something. He was determined not to let this friend of ours push him into doing something that he wasn't really sure he even wanted to do.
And that's from a guy that actually likes anime. If he didn't it would probably have turned him off of it forever.
So, don't nag.
Your best case scenario looks something like this.
Tell your friend in advance that you want to introduce them to anime. They might put up a fuss. Just tell them, "don't worry, remember (insert movie here)? It's a lot like that, and I know you liked that." You can also make promises of having plenty of popcorn and snacks (perfect chance to introduce them to pocky).
When they finally agree, set a date and time. Don't just surprise them with it one day.
Make your show selection carefully. Depending on the person in question, you might only get one shot at this. Don't pick something that takes a few episodes to wind up. They might not want to keep watching that long. They're not going to be as forgiving of a show as you might be, so choose something that starts out strong and stays interesting throughout.
Be sure to be prepared for a binge watching session, because if things go well you could end up going through the whole series (or at least a large chunk of it).
You should be able to tell pretty quickly if they're actually interested or if they're getting bored. If you suspect boredom, don't force them to keep watching. You might be able to salvage things by switching to a different show if you spot it quickly enough, but don't try to force them to keep watching. That's probably the quickest way to completely stomp out any interest they may have had.
If, after it's all over, they seem confused or unsure don't respond with "well you should probably read the manga." You can introduce manga at a later date. This ties in with picking the right anime. You shouldn't pick one that ends on a confusing or uncertain note that wouldn't make sense if you hadn't read the manga.
Try to end on a positive note, if at all possible.
If everything goes well, you may just have a new anime watching buddy. Be sure to tell them where they can find more anime online and encourage them to keep watching on their own. Maybe schedule another anime watching session and have them pick the show this time. Just don't rush them, they're still new to all of this. It'll probably be a while before they're ready for Subs and manga.
Keep in mind that some people just aren't going to like anime no matter what you do. Don't try to force them and don't nag them. Maybe some day they'll change their tune.
