This mostly happens with players below 1300s...When they make mistake they spam draw offers.... happens to me a LOT
Opponent is losing badly... and offers a draw?!

Enormous_Gastrood, How did you quote so many different comments at the same time?
I remember one game in which we were equal on material (both very low), but he was positioned a little better, and I offered a draw, to which he replied, "Oh, I don't think so. I am however accepting resignations today." We both got a good laugh, and continued the game as I said, "Well, I guess you're just going to have to kill me then, cuz I'm a Chess Spartan, and we don't resign!!" We laughed some more, and he proceeded to remove my final pieces until I had nothing left but my King. I said, "I'm going to name my King, 'Forrest.' Run Forrest Run!!" But he could only run for a few minutes before there was nowhere left to run.
The moral of the story is: it doesn't matter if you win, lose, or draw, it doesn't matter if you resign or checkmate, what matters is that you laugh while you're doing it!! Of all the things in this world to get upset over, an online chess game with a total stranger is not one of them.
Brilliant post. Makes me want to buy the platinum membership. (which I will)
It seems to me that getting irritated over a last ditch effort to avoid a loss is grade school stuff.
Don't sweat the small stuff I think. Life is too short.

I offer a resignation by both parties if im losing... that way its not really a draw, we both win a little and lose a little... you know, kinda like a tie
I never resign unless hopelessly lost, but I try not to be ridiculous. I never offer a draw at all. If clearly a draw I will accept an offer if I respect that the opponent has proven the technique to force it. Otherwise, I will not accept a draw and rather risk losing and learn from the experience. A quality loss is worth more to me than a cheap, scared, or safe draw.

I either accept or reject the draw offer depending on whether or not I feel it's to my advantage. If I reject and they suddenly become talkative about stupid things I disable chat. Chess is just about the only thing in life in which I am ruthless. Kick'em when they're down. I never feel sorry for anybody in a chess game. No reason to. It's just a game.
I offer this as an example of the mentality of at least one such repetetive draw offerer...
In my first foray into 'live chess', on Yahoo games several months back, I was significantly
ahead on material and in no time trouble (with both of us having about 20 minutes left on
our clocks) when my opponent started offering a draw after every move.
I just quietly continued moving, until my opponent sent me a message saying,
'Enjoy sitting there for 20 minutes' and just stopped moving......
Unfortunately this tale has a sad ending. I dutifully sat there, pondering human nature, for
about 15 minutes. Then my link went down and I lost by default!
Doubtless he or she still looks back fondly on their victory (dare I say, much like 'rich' with his cherished "victory over a 2000+ player" - http://www.chess.com/forum/view/general/me-vs-chessgirls89)

You might respond: "Seeing some slight pull remains in this position, I must pursue it and hope for Karpovian accuracy to bring home the point..."!?!

SIMON_P ...Wow what an amazing example of poor sportsmanship in that link. After seeing the remarks of the "winner", like below, it's good to know it was an online game so he couldn't hit her with a chair ...
*** *** ***
United Kingdom
"IT DOESN'T MATTER HOW YOU WIN, AS LONG AS YOU WIN DUH."

In a game not so long back I felt that it wasn't going anywhere and offered a draw which was rejected. So I played on only for my position to improve with winning chances. My opponent then asked for a draw which I then refused and I went on to win the game. My opponent on losing became very abusive (sad git). The moral is don't be too quick to offer or accept a draw. Best to fight on until it is patently obvious that you are either going to win or lose

Enormous_Gastrood, How did you quote so many different comments at the same time?
yeah i know huh this is just a waste of forum space
This mostly happens with players below 1300s...When they make mistake they spam draw offers.... happens to me a LOT
I offer this as an example of the mentality of at least one such repetetive draw offerer...mIn my first foray into 'live chess', on Yahoo games several months back, I was significantlymahead on material and in no time trouble (with both of us having about 20 minutes left on our clocks) when my opponent started offering a draw after every move.
I just quietly continued moving, until my opponent sent me a message saying,
'Enjoy sitting there for 20 minutes' and just stopped moving......
Unfortunately this tale has a sad ending. I dutifully sat there, pondering human nature, for about 15 minutes. Then my link went down and I lost by default!
Ignore it, completely.
And take all pieces of his/hers if possible.
Maybe I could promote all of my pawns to knights like Nakamura did against one of his computer opponents.

I've had someone offer me a draw when they were well down on material and close to being mated. I considered this to be poor etiquette, so declined with a reply of 'white would be stupid to accept a draw, black would often resign in this position'. The opponent seemed to be offended, didn't make another move and timed out! It wouldn't surprise me if some people offer draws hoping the winning opponent clicks 'accept' by mistake when it pops up.

"Well you may be right; it looks pretty difficult, but I want to prove I can draw this. Hey you would not believe, one time I won from this position ...what kind of Fish would let THAT happen ??! "
there was this guy who i was playing and he was 300 points less than me.(this was in a real uscf chess tourney). and i was beating him by a rook or something. he said "i propose a draw" and his eyes were moving really rapidly. I was pretty sure he was about to cry and h did. the Td came and took him to his parents. Im sure his parents disqualified him:(

He he its happens all the time..... I dont even answer them... Just make your next move.. Its bad sportmanship to offer a draw when they are a few moves from a bad bad loss.... This people need to stand for what they did and they should send you a "good game pal" and Thanks for teaching me, then resign. NOT a "bad taste in the mouth crap like an draw offer....

I haven't experienced a draw offer from an actually completely losing position, as it portrayed in the original poster's description.
What I have experience though is a draw offer when the opponent has a _slightly_ worse position. For example, each side has King + Rook and equal number of pawns - but the opponent has some tangible weakness (less rook activity, or pawn weaknesses). I usually just say "I'll play on if that's ok with you"

I do this all the time when I am in an obviously lost position. I offer a draw (humorously), then, after a few moments, resign. It is not intended to be an insult; it's just the last desparate tactic to use when all my other tactics have failed. What upsets me, though, is that some opponents have actually accepted the draw offer! This is wrong; I was just making a joke. But there are some people who are offended when you don't accept the draw. I have two words for them, and they aren't "Merry Christmas" or "Happy Birthday."
I had a game once where I was way ahead (a rare event) and my opponent kept offering me a draw; almost every move, he would just offer a draw again and again. amazing.