None, really. I've seen comparisons with Metroid, but those don't feel right either.

There are elements that are similar to other games, but SteamWorld Dig isn't close to being a clone of anything. I already mentioned Boulder Dash in regards to falling rocks (which can be used to get rid of enemies, but it's not mandatory). But unlike in Boulder Dash, removing earth doesn't happen instantly unless you already have a powerful pickaxe; usually it takes several hits to remove a block and the earth keeps getting harder the lower you go, so the upgrades you buy more or less keep you at the same power level as you go (obviously you will plow through the earth in higher areas).

The Metroid comparisons have to come from the upgrades which allow you to actually do something in a few of the early caves, but likening SteamWorld Dig to Metroid is a big stretch.

There's also a good amount of platforming in SteamWorld Dig. A continuous wall jump is a standard ability; which you will need to get back up in the mining area. None of the classic digging games have that. In the caves you have to perform a few tricky jumps; it's mostly about platforming there, not so much digging.