The player would be able to press Shift + a movement key (more below) to “move carefully”. It would take longer (more on that below), but make less noise, lower the chance of setting off a trap, and lower the chance of hurting yourself on environmental hazards such as broken glass in a window frame or rubble.
For the key, it could either be the above (shift + movement key), or toggle-able like safe mode. Maybe % could toggle it on and off.
As for the cost, it would either be a straight-up multiplier on terrain’s normal cost (maybe 1.5x or 2x), or a percentage based on your Dexterity. Maybe start it at 200% for 10 Dex (220 for 8 Dex), and lower it by 10% for each additional point past 10, to a maximum of 150% at 15 Dex. The numbers here are very flexible, I’m just trying to establish a kind of framework.