Excellent question.  I think the answer is directly proportional to whether or not it is a hobby and how strong your OCD is! 

I'd suggest if you want the lowest time investment, point your miner to something like MiningPoolHub (or others) where you have a selection of coins, profit switching opportunities, and all wallets are in one spot (my est 1-2 hours/week to monitor/restart rig when it gets stuck and to count your money!). 

The next level of time investment is setting up core wallets for specific coins... this takes a little more time to manage not just the wallets but the mining software.  (my est 2-5 hours/wk).  Other tools (like Awesome Miner) help with the mining software management and monitoring.  It takes a little time to set it up, but once you have it configured my est is 2-3 hours/wk.

I personally put in a lot more time than that because I find it fun and always find something to tweak! :-)