This is a problem because with 6.5 billion people we're using up resources. Fast.
Anyway, I see two potential solutions to this problem: one not-so-far-fetched and one incredibly far-fetched.
The not-so-far-fetched version is... hitch-hiking. Our understanding of the Universe is that it is full of mass wandering between stars. IIRC recently a planet-sized body was detected traveling between the stars.
Simple probability dictates that it is only a matter of time before a sufficiently large body travels through the Solar System with sufficient velocity (including direction) to reach somewhere else in sufficient time (but not too fast that we can't perform an orbital intercept) that we can essentially build a colony on it.
The far-fetched version is to use back holes as power sources [1] as this is, as far as I've read anyway, the only remotely viable method of providing propulsion without reaction mass to speak of and reaction mass is the death of any form of interstellar propulsion.
The answer to the Fermi Paradox [2] may simply be that it's too hard to leave our comfortable gravity wells and most (all?) civilizations simply run out of stuff before they get there.
I've also given the thought to "footprints". If you think about, say, a primitive tribesman. What do they need to survive in a sustainable fashion? They need a sufficient sized population (measured in the hundreds or low thousands) to avoid inbreeding and sufficient land area to provide a food source. This is probably measured in the tens or hundreds of square miles.
Imagine all they need as the footprint of a sustainable colony as that then dictates the minimum size of any spaceship.
Now imagine a more advanced society. 1000 modern humans would need an ENORMOUS footprint. Just think about computer chips. On any long voyage they'd break down so you need to be able to make new ones. That means a sufficient lab, technology, materials (or, in reality, the means to get more materials), all that knowledge and so on. Plus the size of the population goes up given required specializations.
This of course assumes that people would do all of these things instead of, say, an AI of some kind (which would actually solve a lot of problems).
That footprint is currently way too large to build any kind of interstellar vessel (IMHO). One of the trends I see in coming centuries is that footprint will reach a point of reducing in size. By 2100 I expect we'll be able to keep the sum of all human knowledge (or a close enough approximation) on an essentially mobile device. Advanced manufacturing techniques and materials may solve many of the footprint problems and so on.
As the footprint goes down, the viability of any isolated colony being able to survive increases.
The hitchhiking idea would also be ideal for the survival of humanity overall. With sufficient isolation, there will be cultural and genetic drift. If we're able to influence each other, that's a recipe for conflict. But a large mass passing through the Solar System is very likely a one-way ticket. There's no way to follow and no way to return (barring astronomically small odds of a repeat or inverse body).