No, its not right at all. But I'd like to clarify a few things here.
1) Its youTube. If you want to find something insulting and vulgar, its there. I've seen videos of American soldiers throwing flash grenades at Iraqi shepards as they drive by for fun, or running Iraqis of the road for sport. Does that mean our soldiers are disgraceful, dishonorable fratboys? No. Its means a few of our soldiers with a camera and an internet connection are disgraceful, dishonorable fratboys. Just because some democrats made fun of a baby, doesn't reflect at all on democrats as a whole. If you look, i'm sure you can find something on youtube with republicans making fun of Obama's race - I'm certain that such a video, or even videos, would say nothing about the morality or views of Republicans as a whole.
2) I've heard some Republicans saying this too. Its not a Democrat - Republican issue, its a sexism issue. Nobody would ask this question of a male VP candidate, and if they did, nobody would get upset with the people asking the question. Athough Palin, in choosing to emphasize her motherhood, somewhat brought this type of question down apon herself, its still inappropriate. Once again, though, just because a Democrat says it, doesn't mean Democrats think it. In addition, just because people debate something, doesn't necessarily mean that they believe it. But lets be fair - if you use your down syndrome baby as a political talking point, you have to accept people asking political questions about it. The fact that the McCain campaign did use her baby as a talking point, I think, is a far more relevant topic of discussion.
3) You need to be REALLY careful about saying things like "Democrats are pushing a big rumor." The rumor you're talking about comes from a viral e-mail which is being passed from person to person on the web. No one knows the author, and its not circulating just within Democratic circles. There is no evidence to attribute it to the Democrats, more or less that they are "pushing the issue". In fact, both Obama and Biden have publicly called on the media to not cover the issue on TV, and denounced using children in political campaigning. As Biden put it, "children are off limits." I'm not sure why you would claim that Democrats are pushing this line of discussion, but I can attest that they most certainly are not. Its a sad fact of American society that the most immature and macabre idiocy tends to find a life of its own on the media and on the news. But to use such behavior to charecterize the Democratic party? By the same logic, one could say that Republicans are pushing the lie that Obama is a Muslim, that he isn't actually a US Citizen, or that he is a Kenyan Citizen who regularly intervenes in Kenyen politics. All three are completely untrue, widely believed rumors spread by anonymous or falsely attributed viral e-mails. Simply because they say false and disgraceful things about a candidate does not reflect in any way upon the moral character of that candidate's opponent, more or less their opponent's party. By jumping to such a conclusion, even on anecdotal evidence, you risk becoming a party to a circle of slander and mudslinging. Watch out.
As a blue dog Democrat who knows a lot of Democrats, I can say pretty certainly that I have never encountered any Democrat who would be anything but disgusted by such behavior. Its pure juvenile mockery, and isn't reserved to, or typical of, any one party. It just shows that there are jerks everywhere.