First of all, despite how those minigames are commonly called, the miner and botanist minigames have nothing to do with the miner and botanist jobs in the game at all, except that the way you play those minigames is how you gathered materials from nodes with those jobs back in FFXIV 1.X, before the game was completely remade in 2012/2013.
You do not need to have miner or botanist leveled at all to play these games, nor do you gain any advantage by leveling them.
Also, there is nothing to level at those minigames. You just go there, start the bot and Miqo will do the rest. The different levels of the minigame only change how much MGP you can win, though the higher the difficulty, the harder it gets to win at all. From my experience, playing it at the medium difficulty earned me the most MGP/hour so far, so I recommend choosing that at any time.