I print at 600dpi using a laser printer (4 color) on a gloss heavy weight paper (photo quality). I also make sure that clients realize that print colors are not the same as screen colors so I provide them with a url of screen shots. The other factor is the scren shot program you use. I never enlarge the picture, in fact, I generally keep them small - big enough for most details, but small enough not to degrade. But the key is the printing dpi and a quality printer.

Marshall

You need to scale the image up in pixels, changing just the DPI won't do anything. If you have a 300x300 image and chnge the DPI from 72 to 300 you still have the exact same 300x300 image. The DPI sets the default scale for the image. Import one into Word and you'll see what I mean.

Scale the image in your image editing software in inches to your print size and set the DPI at 300. Make sure to use bi-cubic resizing. This will be pretty good. One thing to note is there is lot of things that can cuase problems, make sure to save to a lossless format like BMP or TIFF. The printer settings can of course affect it too.

When you scale up it produces a softer image but you won't have the pixealtion. You can also scale it in steps and use a noise filtering application like Neat Image in between steps which will give you much sharper image in the end.

If you got a lot of dough and already have Photoshop look up Genuine Fractals.

Almost every screen shot you've seen in print has been modified in Photoshop and cleaned up first.

An image (screenshot) which is 1024 px in width is not much to work with.
At 300 ppi it will be about 3" width in print.
At 150 ppi it will be about 6" width in print.

Sorry if this seems too off-topic - but I'd suggest that taking a portable PC might be a much better option these days, with either the actual site - or an offline demo version (or both!).

Screen prints can be impressive - but if there's just one unanticipated question, they cannot help.

one USB memory stick with the screenshots on it is way better.