General

PAHAHAHA good luck, you will never change her views on this. I've drawn since I remember (kindergarten at least), I'm 26 and still draw occasionally and God I HATE 99% of my art. I've always been praised but I know where all the flaws are, that the lines didn't come out fluently and without mistakes, how much struggle was involved and how I simply gave up on some lines/parts and left them flawed to finish the picture.
Drawing is fun, but to me personally it's also stressful.
I can never get on paper 1:1 what's in my head, that means I partially failed to deliver what's in me to the outside so it was insufficient and I am displeased with it.
Other artists may feel differently but in the end we're too aware of the parts we got wrong than those we got right so we can't learn to love our art.
Which is a good thing, otherwise there would be no improvement. Passion and inspiration keeps the hobby alive, if that dies though it won't feel much like a loss.

Just let her be a and keep praising her, it's sometimes the only thing that keeps one trying again

Edit: Also, while I do like that picture that she made I also sense laziness/insecurity in it. But then again it's only based on this one picture, maybe her other art is more detailed/light/cheerful.
Having a career in arts is really hard and not something I'd personally suggest. She may be good (and I couldn't do what she did but I also don't have a graphic tablet (yet)) but there's many that are better that don't make it. A picture like that may only take a few minutes for some people after all...

I agree with @xDracius by the way, she should keep a log of her drawings (uploading or otherwise putting it in an archive) to see her progress. I probably would have given up drawing long ago if I wouldn't know that I actually can improve even though it takes a while and I usually feel like nothing's changing (compared to 15 years ago though I progressed a lot xD)
other ways to boost her self-confidence about it are
A) upload it on forums and the like to get praise from 'strangers' (and possibly tips to improve too, sometimes it can be frustrating but it can also be enlightening enabling you to draw much better next time (for example when someone tells you about ratios of anatomy you can go from so-so to incredible in a short time. Once I realized that elbows are about waist-level my drawings became more realistic lmao)
B) sell it. It may backfire but she could try Etsy or even ebay though ebay isn't as successful I'd think (and I've seen some terrible items there ... someone trying to sell a terrible drawing for 10$ lol).
She could do prints or try to put her art on items otherwise (like making phone cases), depending on if she's a crafty person.

tell her to stop being a wussy and face the world and all of its criticism that will come with it

if you're trying to encourage her to go to college for art then maybe it's best you stay quiet. i'm not kidding. you'll end up in [url=http://www.basilmarket.com/forum/2886689/0/Tell_me_your_Tifu_stories_D.html]this[/url]tifu thread for causing your sister to get a large debt and a degree that will be completely useless.

May I ask what kinda of art career she plans on pursuing? I'm just finishing up an BFA-Animation program and maybe I can provide some motivation or advice. Sometimes you need a set goal to motivate someone to do something if indefinite goals don't work.

EDIT: I've had no problems finding jobs related to my field so far, so don't always believe the "arts degree is useless" phrase. It really depends on what kind of arts you go onto and what skills you decide to work on. In fact, animators are constantly in high demand, and good ones are actually really hard to come across.

Tell her to upload it for the sake of looking at progress.
She might tell herself it's crap now, but it only gets better from there.

Always nice to look back and actually see how you've progressed, and know that you have the ability to improve because you've done it.

ok if she doesn't want to pursue art then don't make her pursue art, there is nothing worse than someone who is already not confident in their skill trying to enter a field where they'll learn to hate everything they do because criticism is multiplied 200% and the fun will be sucked out of everything. As an art major I can vouch for this. don't push her into something she may regret.

[quote=jonbadspelling]Basically she's been drawing for a while now, she really started getting interested in it back when she was ten. Her earliest drawings are on my page. Looking back on them she thinks they were pretty bad, but my family gave her a lot of praise regardless of how she felt about them. Fast forward a few years and she's gotten remarkably better. However, she refuses to upload most of her stuff to her deviantart account because they're not perfect in her eyes. I disagree completely, but she doesn't seem to care. Anyway, I don't want her to get burned out because art is obviously a career she wants to pursue, and I'm wondering exactly how I would go about encouraging her in a meaningful way. I don't think familial praise really motivates her anymore.

Here's an example of the kind of stuff she's been drawing lately: http://i.imgur.com/JWhUTdH.png

I seriously think that piece of art is phenomenal. If she decides to go to college, I'm hoping that she'll major in the arts.[/quote]

She's being way too hard on herself. I'm the same and i've been told myself that I am too hard on myself, try get her to realise that - she needs to accept that although it may not look perfect in her eyes it looks absolutely amazing to someone else and she has improved drastically.

start a secret patreon account for your sister's art and then after a while if she gets cash (which she probably will if her art is really good) show her the page and shock her with how many people support her art

I don't really have an eye for critiquing things but I'll take it into account as she progresses in her projects. Probably a good idea for her to analyze flaws so she gets used to it in the future.

Wat is that artwork meant to be?

She's an amazing artist but if she can't take criticism, then she isn't ready to be a real artist

Maybe you need to take the opposite approach: get your sister to be more receptive to criticism. I used to feel the same way, I was always protective of my art because it wasn't finished. But the most valuable time to get criticism is when your work isn't finished, so you still have time to make it better with the feedback you get.