I've done this numerous times in my print designs. Many clients want a screen grab of their website to showcase on a brochure or some sort of collateral and want me to make it happen and look good. The method I use is not ideal, but seems to work satisfactorily...
Take a screen grab of the website and bring it into Photoshop.
Create a separate new document that's the final print size of your brochure and, eh, I don't know, 150ppi or so.
Drag your screen grab into that document window and observe its size. Is it the size that you would like it in your final composition? A little too small? Create another new document with the same final print dimensions and change the ppi to 100. Then drag the original screen grab over into that window. See how the grab is now physically a bit bigger on the page, yet it didn't get any fuzzier? That's because the document's resolution has changed, but the original screen grab has not been resized... You just dragged the original image (which was probably grabbed at 72ppi) into a document window which was a different resolution, so it appears to be a different size but the pixels that make up the image have not been altered.
This will give you the sharpest possible results from a screen grab off of the web. If you were to resize your original screen grab to 300ppi, it would look horrible, as you would be increasing the resolution more than 3 times with no pixel information to compensate for that increase. Best to work within the limitations of the original image...
Play with resolutions on the destination document and drag your original screen grab into it until you find a physical size that suits your needs... then, if creating a brochure, I'd recommend converting the image to CMYK and importing it into InDesign or Illustrator and finishing up your type work there.
Hope this helps.