While it's true that everyone's going digital, I've found that face to face meetings with local/regional editors are well-served by a physical portfolio. Following is my personal method for retaining my clips. This works for me, but your needs may be different. Experiment!
I will generally keep one copy of the magazine or weekly that my work appears in, as a whole, for my bookshelf. I'll then take a second copy and actually cut out my article for placement in a three ring binder. This binder acts as my physical portfolio. This physical portfolio comes in handy, as I said, for meetings with locally-based publications. I also do a fair amount of local biz-tech writing, and these clients like to see a physical portfolio, even if it does consist of magazine articles in addition to more targeted biz-tech samples. Last, but certainly not least, I'll scan each article into a PDF and pop it onto my web site. However, carefully consult your contract/Letter of Agreement before you do this, and/or ask your editor for permission! If you don't have permission to publish your clip on your site, that's ok! Make a note of the clip, link to the magazine's website, and indicate that the sample is available via email per request.
Although I think that a web site is necessary for most writers beyond the beginning stages, you may want to start with an online portfolio service, which will store your work in PDF or JPEG form, along with a few biographical lines. This is a great first step to take until you are ready to commit to the complexities of a web site.
My opinion is that your portfolio, whether online or digital, should consist of your very best work. Until you're established, this may mean every single piece of your work, and, depending on storage issues, that's probably ok. However, intermediate and advanced writers will need to pick and choose pieces that best showcase their voice and strengths.