I've said many times that haggling is a lost art. These days the vast majority of people just don't bother, and those few that do tend to go about it very, very, very badly. I've had less than half a dozen customers in the five years I've been doing business successfully haggle with me. People just don't know how to go about it!
Now there are, of course, many different individuals out there, and not all of them are going to respond the same way and I would, but I think I'm pretty typical as an artist, I've got an ego, I want to make what my time is worth, and I want to get as many paying projects as possible. Many artists aren't going to accept haggling no matter what you do, but these guidelines should help with those who do, and hopefully avoid pissing off those who don't.
First. Go into the haggling understanding what haggling is. Haggling is an attempt to persuade the artist to accept less money for their work. Persuade is the key word here. You need to give them a reason to accept your offer. So let's talk about persuasion and reasons.
Persuasion requires that the artist be in a good mood towards you! They're not going to accept less money to deal with somebody they dislike, they're more likely to demand more! Don't make the artist hate you!
So if you take nothing else away from this, for the love of fuzzy things do NOT insult the artist's work or prices! Do not say "You're way too expensive" or "Your work isn't worth what you're charging" or anything that could in any way be construed to mean something like that! This is the most common attempt to haggle I get, and it STINKS. "Holy crap that's so expensive!" isn't much better. Nor is "I don't think X item should ever cost more than Y amount." Think about this! You're trying to put them in a mood to accept less money than they normally take. How is telling them they're Doing It Wrong going to put them in any sort of good mood at all?
Also do not compare them to other artists! "So and so does this for X amount." That's nice, why aren't you hiring so and so then? They're either not taking commissions, in which case their prices are irrelevant, or they're in some way inferior to the artist you're trying to hire, in which case their prices are, again, irrelevant. Or maybe you don't like them personally and don't want to do business with them. Well then, their prices are... guess what? Yup, irrelevant!
If you do want to bring another artist into it, here's pretty much the only polite way to do it. (And it's one of the most successful methods, imho.) "I'm trying to decide between commissioning you, and commissioning so and so. Their prices are cheaper but I like your work better. Could you maybe give me a small discount? That would make my decision much easier." There. THAT is a good haggle. That has praised them with "I like your work better" and given them incentive to make a deal as well.
Even if you're not considering another arist, buttering up the one you're after is a good idea. Just don't go overboard, you'll come across as insincere. A simple "I really admire your work and would love to own some of it" is plenty. "OMG you're the best artist in the whole world and I love you and want to have your babies, you rock so much I just love you to death you are the best ever!" is kind of creepy. Also, there is a difference between real praise, and passive-aggressive manipulation. "You're a great artist, but I don't have much money, can I get a discount?" is one thing. "You're so good that surely you can do this in just half an hour, so for half an hour's work I'd pay you this much, because you're just that good and I know you can do it" is manipulative and annoying. And so what if I can do it in half an hour, you're not just paying me for that half hour, you're paying for all the time it took to develop the patterns I use and the skill it takes to use them.
Now let's talk about sympathy. Getting an artist's sympathy with a sob story is tempting. Be careful though, because artists hear a lot of sob stories. "My boyfriend is in the hospital, it would just cheer his poor, dying self up if you made him a plush for free." (I seriously had somebody say that to me! Well, I don't remember if they said he was dying or not, but it was WAY over the top, and WTF no, a small discount yes, for free, NEVER.) Sometimes it gets to the point where a sob story i s actively annoying to an artist, they're so sick of hearing them. So don't over do it. One that worked on me in the past: "I'm flying to visit my boyfriend, and the tickets are expensive, so I'm trying to save every penny I can. Can you give me just a small discount on this?" Why yes I can! Once again you've given me a reason without annoying me, so I will be happy to oblige.
But that leads us to the "small discount" bit. Be reasonable. If somebody is asking $100 and you offer them $20, you might as well have not bothered, they WILL turn you down. $50 isn't a very good offer either. $80 is more like it, though I might still reject it, depending on how you ask and how close to the margin that particular item is. $95 is almost guaranteed to be accepted unless it's something that I run right AT the margin (in my case that's most of the cheap stuff like Loonakits and pillows and chibis. They're already as cheap as I can make them, they're just not getting much lower.)
Frankly, if you're making a reasonable offer, you don't need to bother with anything at all but "I have $x amount, would you accept that much?" The buttering up and the sympathy are completely optional. (And yes, that does include a reason! If you only have $x, the artist knows if they turn that down, they won't get your business.)
In the end realize that not all artists will haggle, and not all items can be made any cheaper and still make a profit, so don't be offended if your offer is turned down.
And when it comes right down to it, you have to decide if haggling is going to be worth it. Artists are not machines. The work they turn out is not identical in any case. An artist who is grudgingly doing something on the cheap just may not give you the quality of an artist who is enjoying their work. If money is really tight, and that's what matters, haggle away. But if quality product is what matters, you may just want to pay full price. Or even tip! A little extra can go a long way to getting you the best possible artistic results. :)