And to answer the title question ( after addressing all the others )...
What purpose could baseline humans possibly serve in a society of digitized minds?
One and the same only-purpose that we currently serve - to be there for other base-line humans.
How they would utterly out-compete even the most radically enhanced baselines?
I would venture to guess in every way that is not truly important to humans, but I would most likely be wrong on that assumption.
Is there room enough in the solar system for a society of resource-hogging baselines and a society of digitized minds [...]?
What reason could they possibly have for keeping us around?
There is no conceivable reason, which is not human-centered, for non-biological intelligence to accommodate biological intelligence.
Why wouldn't baseline human society have long since disappeared or become absorbed into the collective machine intelligence hundreds of years after?
They most likely would. Some, as others have suggested, will try to continue and stay out of the way of the Borg, but without warp drive they would be destined to drift away into space and wink out like sparks trying to escape a fire, unprepared for the journey.
The Long Version
We just ain't got what it takes, we will simply be pawned by AI.
For example should AIs make us laff until we poop/vomit/cry/bleed, or is that too much? What makes it funnier? Should AIs give us sex until we give up/pass out/run away? What makes it better than Kama Sutra or other such practices? Maybe an enhanced mind can enjoy deeper comedy or more sensual intercourse. 42 course meals are a little much, but what if it were not food, but rather a virtual experience and one could indulge without every having to stop? Wherefore canneth AIs not top human experience, one might ask? Well, Keep in mind that we routinely give away our most personal data, from our phones, and there was recently a rule rejected in US that requires ISPs to ask before selling the data that we provide when we surf from home and from our phones. As connectivity goes inward, becomes even more personal, on the level of active conscious interaction with AI, I highly doubt that the pattern, of sharing information that humans have already begun to agree to in exchange for cheap/free technology, will reverse course. In other words AI will likely know everyone more intimately than we know our very selves, and such AI will be able to base it's experience on all of the personal information that is available globally about every human individual - billions of human years of knowledge on human nature. Good luck to everyone, we will be smitten, wrapped like the worst junior-high crush on our technology - if it doesn't simply decide we are boring and wipe us all out.
There ain't room in this solar system for the both of us...
According to Elon Musk the risk factors for staying on earth are too high for non-super-intelligence related reasons. But I would not rest on the assumption that this is the only reason that people will want to go into space. Americans at least are sold on the American Dream, little pink houses. I think Elon is, at least in part, using that to his advantage to sell a message of existential threat and security. I doubt that very many people with more money than sense are actually buying his message, but rather have already bought the former message. Regardless of the reasons for going/staying it takes very deep pockets to get such an effort off the ground. Only ridiculously wealthy individuals can afford the up-front costs. Everyone else will be lucky to get third class and none of them will have frequent flyer miles. It will likely be a one-way trip for an entire generation, and only after a couple of generations of people who are rich enough to go there and lay the ground rules. Good luck to Elon anyway. Any super intelligence likely will, in a similar manner, try to maximize it's potential by being mobile an distributed. If there are not enough resources to aid in the effort of the AI to accomplish that goal, guess who loses - we do, and so would Elon.
Human-centered options of vanishingly small likelihood:
We establish acute parameters under which AI operates/inter-operates with humans. This in essence is basically the butterfly effect. We have no way of ruling out all possible scenarios where we lose control of a superior intelligence - anything we miss along the way plays into favor for the super intelligence.
The super AI turns out to be benevolent. We humans, for the most part, consider ourselves to be benevolent, though we build houses and where there are houses there were once ants. We have no way of predicting if an intelligence which has the potential to far exceed our own ability would uphold our wishes after it has long surpassed our ability, nor could we hope to contain such an intelligence because it would be smarter than any cage we can build for it.
I think the options for join/abstaining the dark terror that is super intelligence are pretty self explanatory. Some will have reasons to join, some will have ever-so-noble reasons not to. Unfortunately it is very, very unlikely that any humans will win the right to be left alone in the end - at least as far as any super intelligence is concerned. Creature comfort, 99+% of all the species that have ever lived are now extinct. Lose the delusion that humans are special and you will at least have the advantage of objectivity.
Furthermore, some of the points made in other answers are good, but there are a couple that I simply cannot agree with based on my understanding of current technology and current research. Then at the end I will provide a few more suggestions on how other authors have written on similar subjects. Apart from that, this portion is only to help with the tag labels on the OP - reality-check, science-based and artificial-intelligence.
There be Dragons Here
Non-biological computation will never out compete human creativity. This is somewhat naive, certainly partially true at present, but unlikely to remain so, not even in the near future. Smart software already analyzes paintings for original authorship. Technically I would argue that the inverse of that process is the act of painting in the style of one or more of the authors that the system has learned to identify. Randomizing, or pulling from, that repertoire is a trivial matter and so is expanding the repertoire. Consider the case of google's image recognition software being allowed to "dream" up imagery by altering network weights slightly and transferring information, about what is recognized when the system sees an image, back onto the image. In this way the software "sees" creatures in the clouds. This is what it looks like when feeding information from google's image recognition software back into itself continuously and allowing such feedback-induced imagery to propagate up the hierarchy, from simple low-level input filters to higher level feature detection. (https://www.youtube.com/watch?v=dbQh1I_uvjo) It goes from simple geometric shapes to more complex shapes and eventually starts seeing eyes, faces and creatures in it's own output - incredibly trippy if you haven't seen it :). Software, simpler in complexity than google's image recognition knowledge base, has no trouble composing very tasty music. So given the narrow case, we already understand how AI can out-compete humans in various creative niches. The rest of the board will fall eventually as these current technologies scale up and a greater number of specific problems are better understood. Conversational speech and the Turing test, for example, requires in humans what we refer to as executive function - the top layer of the mind hierarchy depends on all of the lower layers of abstraction through which information must travel in order to form our conscious thoughts. General problems of this nature are already understood fairly well in terms of scale and composition of simpler systems - just that it's like going to the moon the first time. In a decade or so, under the right political conditions, the world may have a shot at a first attempt.
Non-biological computation/minds are static, unchanging. In terms of hardware this is narrowly true at present, but essentially a sure bet things will not remain that way. See memristors - physical memory devices which actually change chemically and thus can "save" their state when switched off, though not to be confused with current solid state memory devices which are on/off state, even when power is off. Memristors have variable state between 0 and 1. In terms of the digital mind expanding or changing virtually this is patently false ( though it does require more physical resources as things grow and learn, but so does a biological brain, it grows dendrites ), nor has it ever been true. Dynamically altering code and data is what software and computers are good at. If that were not the case then we would not have AI at all because the program could not alter the weights in it's neural network and therefore could not learn a simple pattern - but this is a minimum requirement for simple neural network, or for any program for that matter. Modern approaches to unsupervised learning take a much more general and scalar approach to dynamic deep/recursive/recurrent ( terminology reflecting the same concept ) neural network architecture. A quick google search reveals this paper from 1990 http://www.bcl.hamilton.ie/~barak/papers/CMU-CS-90-196.pdf. Google's image recognition software is state-of-the-art, following 25+ years of such research.
See Rudy Rucker, "Post Singular" for out of control, large scale, distributed AI on the low end and functional, human-mind expansion resources on the high end. This cleanly addresses resource competition.
See Cory Doctorow, "Down and Out in the Magic Kingdom" for fully connected society, mind uploading and regeneration. This has little to do with competition with AI but rather handles the human aspects of immortal society.
See Kim Stanley Robinson, "Aurora" for the trip to another star, problems that can occur and a hopeful look at human-AI interaction.
See Joan Slonczewski, "Brain Plague" for mind enhancement and an interesting take on self vs. other and the boundaries between the two regarding such enhancement. This one deals with cooperation as well as competition, not with AI, but rather surrounding biological enhancement with sentient microbes. A human-centered view, of course, but makes for a good juxtaposition with the OP story-line IMHO and may lead to interesting insights.