I'm still testing something fully that came to mind, but I think you dont want to resize "this way" with either tool.
The website is fixed at 950 pixels. So you can adjust the DPI in a couple of places here to make your imports not need to be manipulated as much. There will still be a little bit of aliasing but thats the nature of the beast. This same aliasing is caused even if I zoom the page down in size in the browser. At least in Chrome, Firefox, and IE. So I dont think you're going to get much better with text embedded in an image without using Adobe tools. However this other approach shows a little difference in the output to PDF.
To shrink a 950 image to 6.5 inches for publisher, you would set the DPI to of the canvas before pasting to 146.2 DPI to cause it to fit at 6.5 inches when imported. Technically this isn't much different than letting publisher have its way with the images but it seems to help ever so slightly in my fast testing.
Image of new canvas dialog (height can be adjusted to whatever is needed or matches ...):

Also, save your shots as PNG. This is lossless format and will not lose information when imported into Publisher. You're doing enough pixel crunching without doing any more damage by gif or jpg. They really are not good for this sort of thing.
Exporting to PDF in Publisher: Commercial Printing option. It will flag your doc as having a problem: RGB color space. Once converted to CMYK plus spot in my testing, my PDF's actually shrank in size slightly without fidelity loss and images at 300+ DPI no longer had pronounced aliasing as seen with the High Quality Printing option. Especially if you are going to potentially exceed 150 DPI, you have to choose these options or it will mangled stuff.
This may help you work around the problem and get a slightly better result. I think I see a difference.
No matter what though, when you take a website screen-shot (combinations of images and text), then crunch it into an image, its going to lose some of the fidelity. Especially with the text. This is one area where Adobe is king as it tends to do things that it holds the patents for. I'm not anti-MS by any stretch. Just in the world of PDF's, we all have to seem to pay homage to Adobe ...
I hope that makes sense. Rather hectic day, but I was curious about the issue and thought I might be able to help find something of a solution. So far I'm not sure I've done more than show you a parlor trick to match DPI of a canvas to make resizing unneeded in Publisher for future needs. (Like headers, etc.) I've used that method specifically in Word docs to avoid some sizing issues, etc. Just wasn't sure what impact it may have in Publisher on image quality when compressed to PDF. So far I'm not sure its much different. But I do prefer to use PDN to cut up screen shots.
*Almost forgot: set you selection box tool to a fixed width of 950 pixels by whatever height works for the capture. I have to guess on the height you would want to grab as the site seems to vary a little.
Edited by delpart, 29 February 2012 - 04:50 AM.