I think this depends on the degree of inter-user interaction your community has. At one end of the spectrum, consider a mailing list that tends to focus on solving specific problems, where the content is overwhelmingly what's important. In such a case, the archives are useful to the Internet at large long, long after any of the people who contributed to them are gone. (This is even more true if the content was curated.) In that case, if you can host it, publishing the archives usually makes sense. Google searches have led me to such archives on many occasions, and I trust I'm not alone in that.
At the other end of the spectrum, if your community is primarily about the people, like an online social group, and the content is secondary -- the kind of community where the words "that's off-topic" are never uttered because if Bob wants to talk about it then it belongs here -- then not only is there no point in maintaining a presence but it actually might not be in the members' interests. Despite the fact that anything posted on the Internet should be expected to live forever, there are groups of people who treat their little corner of the net as "family", and such communities may prefer a full shut-down to an archive or a partial migration.
Most communities will be somewhere between these two endpoints.
I have shut down communities that were closer to the "archive" end of the spectrum. In one case I found another community member whom I trusted to continue the community, and I handed it over and remained a member. In another case, interest was clearly waning, I got no takers when I asked for a successor, and I decided to shut it down. I didn't publish archives because they weren't really of lasting interest.
A word about archives and aging:
Sometimes archives are timeless; the information you compiled about cooking, or renaissance dance, or craft brewing is likely to remain about as useful in the future as it is now. Other times, your content loses relevance over time but doesn't become outright wrong; there probably aren't that many people who care about writing programs for Windows 3.1, but if you want to publish it, sure, go ahead. Yet other times, your content might become detrimental over time, such as if you are in a medical field and treatments previously recommended are now known to be harmful. So consider the volatility of your content, and if it's anything other than timeless, consider periodic review and curation. If your content does not have a long shelf-life, or if you aren't willing to do some upkeep, it might be better to take it down even if it's still useful now. At the very least, make sure it is clearly dated.