Little humans are incredible things. We're born with a lot of instincts, many of them actually related to climbing — for instance, if you put your finger into the palm of a newborn baby, they will instinctively grab onto it, even when asleep. This is a throwback to when losing your grip on that tree branch was a fatal mistake! It takes a long time (9 months or so I think, not sure of the exact period) for this instinct to be overridden by a lack of requirement.
Secondly, babies are strong. Anyone who has tried to get a small child arching their back into a car seat can attest to that! Relative to their size and weight they are capable of an awful lot more than it looks like they can.
Thirdly, pain is an instinctive response, it's not something you "learn" to pay attention to. If a cat scratches a baby, the baby will cry and likely be scared of the cat. They don't actively seek out a repetition of the pain. In this case, a baby won't wilfully put itself in a position where it is feeling pain, such as attempting to support its weight through its hands before they are ready. It's only as we get older that we learn to be stupid enough to ignore the messages our body sends us!
So essentially, the act of climbing itself won't cause your child any harm — If they are strong enough to climb, their body is strong enough to take it.
There are, however, caveats (as always). While the act of climbing itself isn't harmful, the results of climbing can be:
- If the baby slips, and jolts its arms by instinctively grabbing on (falling reflex), it could theoretically get something like Pulled Elbow. This kind of injury doesn't do lasting damage, but it is sore! It's like dislocating a joint.
- The higher your baby climbs, the more like it is to hurt itself if it slips. It's true that babies bounce, but this isn't something to be tested :P
- Attempting to climb a bookshelf (for instance) can lead to things being pulled over on top of them. THIS IS VERY DANGEROUS. If your child is inclined towards climbing, make sure everything climbable in their reach is fixed so that it won't fall over.
- As stated in the comments, if there is something they can climb, make sure it doesn't lead to somewhere dangerous. If you think they could climb your stair gate, you need to change it/modify it so they can't. A stair gate is pretty useless if they can just climb over it!
Overall, it's something to be encouraged in the right environments (places that are safe to climb, even consider taking them to a climbing wall (if you're an avid climber then presumably you have places to climb nearby! There are climbing gyms near us that have bouldering areas and similar for toddlers and babies), and it's something to discourage in the wrong environments — like many other things, there are certain lessons that can't be learnt through experience. You just have to be consistent about saying no to climbing the wrong things (like stair gates!).