Careful is the better choice in this context.
Cautious and careful are synonyms, but imply different things. With both, you are taking care not to do anything wrong, but with 'cautious' there is more the sense you are anxious about making a mistake or fear the consequences of making a mistake.
For instance, if I said I approached the edge of the cliff 'cautiously', there is a strong hint I fear making a mistake and falling. If I said I approached the cliff edge 'carefully', I am only saying I do not want to make a mistake. I am not saying so much that I fear the consequence of making a mistake.