Under "Noticing Threats" (PHB p.182) :
Use the passive Wisdom (Perception) scores of the characters to determine whether anyone in the group notices a hidden threat. [...] While traveling at a fast pace, characters take a -5 penalty to their passive Wisdom (Perception) scores to notice hidden threats.
When exploring the dungeon, passive perception is what would normally be used to detect threats. In this case, passive perception represents your character's habit of always looking over his shoulders, of checking every nook and cranny and every tunnel he encounters.
If you state that you are taking extra precautions, an active roll is definitely possible. When I DM, if the PCs tell me they wish to proceed cautiously, I use what I have dubbed the "reverse sneak rule". When sneaking, you are normally limited to a slow pace, because you are being very careful not to be seen or heard. It seems logical to me that applying the same level of care to spot hidden traps and sneaking enemies would require you to move at such a pace as well, as you are then looking behind every tree and in every shadow, stopping frequently to hear for approaching enemies, etc.
Of course, your DM can feel differently about it and always ask you to roll perception checks, which is also fine!