In a dungeon crawl adventure, the characters have different goals: finding treasure, not killing monsters. In fact, if you can get through the dungeon with minimal risk, so much the better, because the payoff should be the treasure. Combat serves roughly the same purpose as traps: an obstacle preventing the characters from getting to their goal.
Older editions of Dungeons & Dragons (and later retroclones) solved for this by rewarding experience for treasure, not defeating monsters. As an example: 1 experience point per gold piece recovered from the dungeon. Gold gathered in other ways (e.g. business ventures in town, robbing shopkeepers, etc.) didn’t count. Some variants required the players to spend the gold used for levelling on something relevant: training, carousing, donations to an appropriate temple, or anything else appropriate to the character.
In 5e, this still works, especially in a dungeon-focused campaign where the characters will find adventure and fortune climbing down into a dark hole in the ground filled with monsters and unknown dangers. Other types of exploration, like marching off into the wilderness, work too! I wouldn’t suggest this for all campaign types, certainly. This sort of fundamental change to the rules requires building the campaign around it.
Originally, I’d thought about doing this instead of combat XP, or maybe reducing combat XP to some fraction of the normal value. But we want to increase choices, not reduce them. “Experience for gold” gives the DM more knobs to turn for managing advancement, assuming they want to use something other than various forms of milestone XP. Most DMs award XP just for monster encounters, combat or otherwise. The Dungeon Master’s Guide gives very little advice for encounter-based RP otherwise:
You decide whether to award experience to characters for overcoming challenges outside combat. If the adventurers complete a tense negotiation with a baron, forge a trade agreement with a clan of surly dwarves, or successfully navigate the Chasm of Doom, you might decide that they deserve an XP reward.
As a starting point, use the rules for building combat encounters in chapter 3 to gauge the difficulty of the challenge. Then award the characters XP as if it had been a combat encounter of the same difficulty, but only if the encounter involved a meaningful risk of failure.
Instead, tie those non-combat challenges to exploration, and reward treasure for them. The random treasure tables in chapter 7 of the DMG provide much more specific and quantified guidance than those two paragraphs above.
I just had a thought. Maybe we could tie this to downtime activities, including the complications listed in Xanathar’s Guide to Everything? That requires a bit more thought to sew it all together, but I like the idea of giving more texture to the time between sessions. This already covers carousing, religious service, training, research, and more. In a West Marches campaign, custom downtime activities to improve the town…
Maybe the scope has grown a little, but the core idea remains the same: implement sinks to give the players something to spend their treasure on and get experience in exchange.