This question was closed as a duplicate of another question. However, I believe it is not.
The linked "duplicate" question is system-agnostic and is good for general advice. However, the closed question is specific to D&D 5e, and 5e has specific mechanics for handling the question.
First off, the DM is missing the point on skill checks, because it is called out specifically that "The DM calls for an ability check when a character ar monster attempts an action (other than an attack) that has a chance of failure." The DM calls for checks, the players don't.
Secondly, "I am always vigilant" is handled with passive checks: if the DC is greater than their passive check, they don't notice whatever it is. Increasing the ability score (Wisdom or Intelligence) is one way of raising their passive score, but there are also a handful of feats that provide a much larger improvement to passive scores:
- The Alert feat means they can't be surprised
- Dungeon Delver grants advantage when looking for secret doors and when saving vs. traps
- Observant grants a flat +5 to passive Wisdom (Perception) and passive Intelligence (Investigation) scores—among other benefits
I believe these three feats cover what the questioner's players are asking to do:
player: I check for traps and secret doors in the wall, the floor and the ceiling. I also want to hear anything unusual and walk silently.
Traps & hearing anything unusual are covered by observant. Secret doors are covered by observant and dungeon delver. When observant isn't enough to spot the hidden threat, dungeon delver helps mitigate traps, while alert helps to mitigate hidden foes.
Therefor, I believe the linked question is related, but not a duplicate since there is a system-specific answer to the question.