Either space is compact, or it is infinite. If space is infinite in extent, either it is thermal over an infinite volume, or it is in the vacuum state for most of it. If it is thermal, infinity is a large place, and random statistical fluctuations alone will ensure there are infinitely many places in the universe just like the Earth (or maybe just your room, or your brain), but not necessarily beyond. This is Max Tegmark's level I multiverse. Only a doubly exponentially small fraction of the universe will be like that. Doubly because human length scales are already exponential compared to the microscopic scale, so it's exponential of exponential. But infinity is a large place. If most of the spatial extent of the universe is in the vacuum state, in quantum field theory, we have the Reeh-Schlieder theorem for any bounded region, and so, there is still a doubly exponentially small probability that the region will be just like your brain. If space is compact, either it remains bounded in size over all time, or it expands indefinitely, or it will end in a big crunch. If it is finite and bounded, it will undergo Poincare recurrences, and there will be infinitely many brains just like yours spread out over time. If it expands without bound, eventually the volume of space will exceed doubly exponential, and there will be many replicas of your brain spread out over space in the future. So why aren't we Boltzmann brains? In general relativity, two locally identical Boltzmann brains can be related by a diffeomorphism. There is no absolute position or absolute time distinguishing them. So we have to identify all these Boltzmann brain replicas relationally. Are they all the same brain, all those infinitely many of them? Infinitely many identical brains, all identified. Quantum evolution is nondeterministic, so they don't all evolve the same. So figure out the fraction of identical brains evolving in a certain manner, and we get frequentist probabilities. This requires a measure over these brains. There is no invariant measure in eternal inflation, the measure problem.
There are three different kinds of brains to consider here: Boltzmann brains, evolved biological brains, and simulated brains.
There might be infinitely many Boltzmann brains spread out throughout spacetime, but what if there are also infinitely many evolved biological brains? Presumably, these infinitely many evolved biological brains are spread out over space, and not so much over time. If they were spread out over time for all infinity, and we're an evolved biological brain, wouldn't we observe the big bang occurring infinitely long ago? So, if time doesn't end after a finite period everywhere in a big crunch, Boltzmann brains will still dominate over evolved biological brains by an infinite factor, even if there are infinitely many evolved biological brains spread out over infinite space.
However, what about simulated brains? If a technological civilization evolves which lasts forever, with infinitely many ancestral simulations over all time, as spelled out by Nick Bostrom, can they dominate over Boltzmann brains? If space if infinite, or finite at any given moment, but expands in size without bound over time, this can only be so either if there are infinitely many technological civilizations, or a single (or finitely many) technological civilization starts colonizing more and more of space and keep growing in size so that asymptotically, it keeps occupying a constant fraction of the total space of the universe for all times.
This is a strong argument in favor of the simulation hypothesis with civilization lasting forever which keeps expanding without bound.
One different conclusion that derives from a previous answer would be the following.
If the universe is infinite enough to reach thermodynamic equilibrium then Boltzmann brains would largely surpass the number of biological brains, this is simply because entropy fluctuations will favor them (they would require a smaller, and thus more likely, departure from the black hole filled thermal/entropic equilibrium state).
But if the universe ends up in a big rip, big crunch, or whatever, then the universe would never live for an enough time to let these fluctuations appear (and then biological brains would be more likely than BB's).
Using the anthropic principle, this would suggest that the number of universes in which their dynamical laws allow complex structures to appear relatively easily, might all be short lived.