What would mankind use in order to adapt or merely cope with the effects of a large increase of gravity on an alien world?
Assuming there are no other problems we would face.
This is not only relevant to the off chance, distant future human colonisation or otherwise long term stay on celestial bodies with larger than Earth surface gravity, but also constant acceleration space travel at over 1 g. It's a fair question, and to be fair in return, as far as adaptation to hypergravity conditions goes, we haven't a clue to what degree it could still be considered safe for humans and at what point you'd start taking risks of increased heart failure rates, respiratory problems, dizziness and inability to perform basic functions as the inner ear fails and we'd become disoriented, or even experience hypoxia or equally deadly nitrogen narcosis.
Some studies of mammals adapting (or not as they have shown) have been done with centrifuges. For example, NASA Ames et al. conducted a study on rats (PDF) that concluded:
Exposing rats to hypergravity via chronic centrifugation resulted in an acute gravity dose dependent decrease in both core body temperature, and gross locomotor activity. Circadian rhythm phase and amplitude were altered by the exposure to the hyperdynamic conditions on the centrifuge.
But these were rather short-term (days or weeks) experiments on four-legged critters with small bodies. Not the best representatives for what problems humans would face in hypergravity at all.
My take is that natural adaptation of humans to hypergravity environment is likely impossible in environments with largely increased gravity because we're simply too tall and our cardiovascular and respiratory systems wouldn't cope for long periods of standing up and doing things as we normally would. I've given away one clue how to partially solve this now, though. Don't stand up. If this could be done for extended periods of time and still permit humans to do much anything to make it worth the trouble, that I don't know, but it's a start.
Another way could be by wearing compression garments, not unlike those that fighter pilots wear while they're expected to perform high gee maneuvers and compress lower body to momentarily cut the blood supply to the legs and increase blood pressure in the rest of the fighter pilot's body, allowing the blood to still reach their brains and they don't fall into what's now known as a G-LOC (G-force induced Loss Of Consciousness). Problem is, compression garments in constant hypergravity environments couldn't deflate for long enough time to let its occupant breathe normally again and allow blood flow through lower body. So they would have to be redesigned to likely inflate (compress) and deflate (decompress) at intervals following one's heartbeat to support human cardiovascular and respiratory function.
All the while, we'd also most likely have to wear rebreathers to support and control our respiratory functions even if the hypergravity planet we'd be inhabiting has perfectly acceptable and breathable atmosphere. I've already mentioned some possible unfavorable scenarios regarding our respiratory system at such conditions, but another one not excluded would be pulmonary edema resulting in asphyxiation simply because of the additional pressure on our lungs and the associated difficulty with breathing. Healthy individuals in their prime years would likely be fine in this regard for a short time, but that's not enough for colonisation, merely perhaps outposts.
So, to recap, we don't know. There, I said it.
Commentary: Perhaps more interesting than the fact that we don't have enough data on adversity of hypergravity on humans is the question why don't we. One immediately apparent problem is the risk involved with doing such experiments and that some of the adverse effects of it might take years to develop and with no way of knowing we'd be able to prevent or at least manage them once they do, since we wouldn't even know what exactly to expect. But there are strictly technical limitations too.
On Earth, centrifuge systems don't really simulate hypergravity conditions precisely enough since there's always a downward facing component (Earth's own gravity) adding to the list of acceleration forces. So the combined acceleration vector in such horizontal systems is actually inclined to the plane of rotation somewhere between 0 and 90° but never one or the other as long as the centrifuge rotates. And if the centrifuge is vertical, combined acceleration forces oscillate on each rotation and result in something akin to a tumble dryer.
To my knowledge, the largest such horizontal centrifuge system ever built and actually put to use was by Soviets during the early human spaceflight endeavours and it was big enough that it could house cosmonauts for days at a time. It used a sloped floor and living quarters area. As you can probably imagine, it was a lot more fun for those running the experiments as it was for their subjects. Acceleration gradient along the radius and rotation alone through Coriolis effect produced some interesting but expected side effects:
Cosmonauts training in Star City centrifuge system demonstrate Coriolis effect on the flight of darts as the centrifuge rotates.
But perhaps more importantly, cosmonauts experienced dizziness, nausea, lost motoric skills due to effects of acceleration / simulated hypergravity gradient affecting the inner ear's and hippocampus' ability to translate sensory information to sense of orientation and ill-affected spatial memory, they would regularly display their lunch after they consumed it (or tried to), and needed days to weeks to recover.
It was by no stretch of imagination a system that could precisely simulate effects of prolonged stay in hypergravity environment. And building even larger such horizontal systems to somewhat remove acceleration gradient, and one with direct access through its focal point as it spins, is out of the question due to technical and material limitations. The last thing you'd want in a centrifuge system is the Galloping Gertie effect. Unless you build a circular track, you're fine with not being able to simply enter and exit the centrifuge system while it operates, and can maintain large enough and constant speed of the carts. Perhaps a circular track maglev?
Alternatively, all this could be solved by building such centrifuge systems in orbit, i.e. space wheels rotating at larger than Earth's surface gravity equivalent centrifugal acceleration, but there's another problem that they would have to be absolutely enormous and with radius large enough to also remove the strong gravity gradient along a human body and approach levels of it small enough that it could be claimed it's largely insignificant and not affecting your experiment's results to a great degree.
Doing this in orbit is however a far cry from our current technical capabilities, because of all the support facilities needed to run such experiments on humans in a controlled environment. Getting test subjects willing to participate might not be easy either since you're by law required to operate under full disclosure (in US and EU at last) yet you wouldn't really know what to expect. So you'd have to be prepared for any eventuality, and that simply means also having too many facilities in orbit that we don't yet have. You'd also run high liability risks of such operations.
All this means that no space agency or research institutions have come even close to setting anything like it in motion, let alone started doing such experiments. It's one of the great unknowns we'll most probably have to deal with by relying on robotic exploration until limitations of our own bodies can be addressed by our technology. We're not there yet.
If you're interested, these issues were also discussed from a bit different perspective (studying effects of reduced gravity and long term presence on Mars on humans) during the recent The Space Show Classroom edition by Dan Adamo, Dr. John Jurist, Dr. Jim Logan and the host Dr. David Livingston. They were mostly commenting on NRC's Pathways to Mars study but they did mention lack of such studies and why that is so. Recording and further info are available here (MP3).
There exists a technology known as "sitting" which is commonly applied in multi-G environments on and near Earth, for example during some airflight. So, the answer is that we'd use our bottoms.
Given the same density, surface gravity increases only linearly with the radius of the planet. And already at a few times earth's radius, planets are thought to collect huge hydrogen atmospheres, like our gas giants, where we'd rather swim than sit. Bare rocks with more than double Earth surface gravity are likely very rare.