If it is a regular stone and mortar wall with no added difficulties, adventurers are deemed capable enough to climb in 5e - they can climb at half speed (unless they otherwise have a climbing speed) with no roll needed.
As evidenced by the Strength ability check section:
Athletics.
Your Strength (Athletics) check covers difficult situations you encounter while climbing, jumping, or swimming. Examples include the following activities:
- You attempt to climb a sheer or slippery cliff, avoid hazards while scaling a wall, or cling to a surface while something is trying to knock you off.
Correct, the DM has stakes in mind for success and failure when calling for an ability check in 5e. Some DMs even share those stakes with the player before the roll to reflect that the PC is a capable adventurer at least partly aware of the possible consequences of what they are attempting.