The Youtube stabilisation question automatically comes up after your video is rendered completely. This can take a few hours after uploading. You can also choose to edit and enhance your clip and you will see the option to stabilise at some point.
However, the youtube stabilisation algorithm is a bit extreme. I rather use my own software before I upload it.
One tip: if you need to stabilise your video your own software, only stabilise the parts that really need it, so cut your video into separate clips, adjusting the stabilisation amount per clip. Sometimes the shaking is so high, the software has to crop a large portion of the image. Be aware that you sometimes have to smoothen transitions from one clip to another using fades/blends.
If you use the same amount for the whole video, it is 'stabilised' even before you take off. Very weird, totally uncool and very not pro.