What drama would you recommend to someone who has never seen any, and more than likely would go into it expecting not to like it?

Depends what is their favourite genre I guess. 

If Mystery/Crime I'd say Special Affairs Team TEN is a good start. I'd generally would pick something that is close to the western production values and don't have cultural nuances.
I agree, it depends on what genre they like.  That said, I think dramas like City Hunter or Healer are are a good mix of genres that seem to have a wide appeal.  I've heard of a lot of people getting hooked with City Hunter as their first drama.
Hmm I think I would start with movies rather than dramas since it's shorter and more likely to draw them into the genre (if there is any interest that is). Then, I would indeed adjust to the taste of the person you want to show the world of asian dramas to. If you don't know, I'd go for a comedy or a highly acclaimed/famous movie. eg. My Sassy Girl, Oldboy, Vengence trilogy, Departures, Rurouni Kenshin,...
Good luck!



Best  K dramas for newbies:

Boys Before Flowers

You're Beautiful

City Hunter

You Who Came From the Stars

The Heirs

Playful Kiss

To the Beautiful You


J dramas:

Hana Yori Dango

Itazura na Kiss Love In Tokyo

3 Peace


Taiwanese dramas:

It Started With a Kiss

Hana Kimi Taiwan


Mainland China:

Say That You Love Me

Boss and Me




I suggest these because they are easy to understand get hooked on, plus do not require a lot of knowledge or interest in history.
What you say is a bit too general. We could recommend you some if we know what your favorite genre are, what country you prefer, etc.

Well, nevertheless I would just recommend you some Korean.

Korean romantic comedy (romcom), they are mostly 16 episodes only:
The Greatest Love
Oh My Ghostess
Marriage Not Dating
My Love From Another Stars
It's Okay That's Love
Producers

Korean family drama, they are usually >50 episodes, a bit slice of life center around family's life:
My Daughter Seo Young
Unexpected You
Wang's Family
What Happens to My Family
birth of the rich
quote from my review "It isn't the best. But for me it has
what is tops about them ie great characters, good story, and a nice balance between laughter and tears.
Agree with needing to know more info on preferences. It might be a bit hard to pin point exactally as nearly 200 dramas down and i've decided i really do like crime dramas after cruel city and bad guys (well worth a watch).

A good comical start to Japanese dramas would be Hana Kimi .. You cannot watch asain dramas and not watch this! it is so very very funny although it is reletively long as just involve cross dressing but its subtle and you forget about it after a while.

A good start for Korean dramas would be something like Boy over flowers. Easy to follow, great ost and hot actors.

A good start for Taiwanese dramas could be just you

And one for Thai without doubt is Full House.

Once you start a few you will start to follow the actors/actresses from those dramas. A few actors and actresses who never let you down .. So Ji Sub, Jung Kyung Ho, Ji Chang Wook, Gong Hyo Jin, Yoon Eun Hye, Kim Sun Ah.

Good Luck
I guess I should have clarified that this isn't for myself.  I am at around 200 dramas watched, and don't plan on stopping anytime soon.  I just always wondered what people recommended to people they knew in an attempt to get them started.

For instance, I got my mother and one of my best friends hooked on Running Man.

My sister is the one I would be at a loss for, as I know she hates reading subtitles.  Only shows I can recall her watching are One Tree Hill and Scandal.  
You Who Came From the Stars
Prime Minister and I
It's Okay, That's Love
Thank You
I Hear Your Voice
Master's Sun
Fated to Love You
I Remember You
Faith
A Gentleman's Dignity
Cunning Single Lady