A., my now 3 year old has always been quite a climber. Before he could actually walk on his own, he figured out how to move items (by crawling and pushing them) to the counter and climb up it. He even figured out how to layer it so he had "steps" if what he had grabbed was not tall enough! Needless to say, we were a bit panicked at this. At one point, I took all the chairs and locked them in a room, so we had to go get chairs to sit down to eat, but he wouldn't stop climbing them!!!
Anyway, for the crib I finally just put the side down on it. It is still up enough for her not to just roll out, but she can get out without seriously hurting herself. It worked for my son. The frustrating part is that it means you have to work on keeping the child in the room, because now they can get out.
There are also crib tents, but they never worked for my son. He figured out how to get them off. And, for me, I preferred that he learn it correctly the first time. This did mean that the room had to be thoroughly baby proofed.
As far as the rest of it, you just have to keep an eye. I found that nothing really worked, because he was bright enough to figure it out. Believe it or not, he even picked the lock into the bathroom after watching me do it! So climbing is most likely not going to be your last issue.
One thing that's important is to make sure you have shelves and other climbable high things strapped to the wall. We ended up putting my son's bureau into his closet and putting locks on the closet, because he climbed the shelves one day (it was safely attached to the wall, so didn't tilt) and got Vaseline down off shelves we had above the bureau. I won't describe THAT mess. :x
I'm trying to think of what else we did, but it was basically us fixing each problem as it came up, because he was often a step ahead of us. I still closed the gates, even though he could climb them, because it at least slowed him down until I could get there. I've heard of buying double the gates and stacking one on top of the other, as well, so that might be worth trying at stairs.
And I found a place that had neat foam toys for him to climb. Not letting the child climb at all is just going to frustrate them.
Good luck!