Around 1 year. Between 9 and 12 months, babies learn to crawl. By 12 months, most are starting to pull up and cruise on objects, and many are walking. Shortly thereafter they start to try to climb things. I always caution parents to start with the "mobile childproofing" around 7 to 9 months to avoid the "early walkers" and the accidents that happen when parents didn't know their children were that mobile.
Once mobile. Children are natural explorers and are fast. If you have stairs gate them off to prevent your child from climbing when you are not watching and with him. Most children can climb up pretty easily but have trouble descending. They need constant supervision to avoid a fall. Anytime after a child learns to walk they may try.....Some before. Children are surprising.