Does the Genie put mana in the bank?

No, your Genie unfortunately does not put extra mana into the bank. When you're capped out, your mana does indeed go to waste.

You can use high level Genies on a character with high DPS to max out towers within 2-3 waves. Try to upgrade as often as possible or use your extra mana to power sustained abilities like Monk aura toggles, Squire buffs, etc., though the mana costs of these skills do rise the longer you keep them on, so this is not permanently sustainable.

You can verify this by looking in the bottom right corner of your screen. Whenever you earn mana that goes towards your banked mana (like at the end of waves for items on the ground, at the end of maps, etc.), you'll see your mana amount change. This doesn't occur during Genie use.