How do people mine minerals from caves?

They do not - apart from a few less-developed areas of the world where extracting speleothems for sale to unsuspecting tourists or unscrupulous dealers is seen as commercially acceptable by the locals trying to live in very poor areas. Caves do not contain economic minerals.

In a few areas, such as the Peak District in the English Midlands, and the Mendip Hills (SW England) 18-19C lead-miners broke into caves by chance, and they did search for lead-ore (galena) there but rarely if ever found useful amounts.

They did however sometimes rob caves of their stalactites to sell to wealthy people to decorate their trendy artificial garden grottoes. One of the worst offenders was the poet Alexander Pope, who hired local marksmen to shoot down stalactites in Wookey Hole for his grotto: God knows how many were simply smashed to useless gravel in the process.

Removing natural objects from caves except for genuine scientific research or to save them from destruction by quarrying, is rightly regarded as wanton vandalism; and is highly illegal in many places.