This is a problem all games designers have had probably since the beginning of designing games. Your main problem is going to be unforeseen interactions and if you make your rule set publicly available people deliberately interpreting rules to their own advantage.
There is a quick answer: you play-test. You play-test a lot!
You discover if your design philosophy or feature set is 'balanced' the same way you discover anything. You have a hypothesis "This class is balanced" and then you run experiments to prove or disprove your hypothesis. You test in game and see if your hypothesis is defensible.
I will run you through what I do when I am designing or developing a new game or class within an existing game. I have 3 stages to this, which I run through in this order:
I make a sketch of the class I am going for and check that in and of itself it is balanced ie seems appropriate for the feel I am looking for, eg do they have a plus that makes them fun while still having a disadvantage that adds long term challenge. Also, just sit with pen and paper and think how can I break this. If you sit and find nothing, try harder, you have almost certainly broken at least 1 thing on your first attempt.
I next run through a number of basic scenarios, New Class of Bob Lv1 vs Goblins, Bob lv 20 vs dragons, etc... keeping the CR rating appropriate if the game I am developing for has such a concept.
This is very basic combat balancing.
In your case of multi classing, you would want to do the same, but then classing a Bob Wizard, a Bob Fighter, a Bob Space Pilot from Mars etc... run them through appropriate scenarios and check that they work as you consider balanced.
- Get anyone you can find to play a character of your new class Bob, first try playing with your usual group and tell them to try break your class. Tell them to try an misinterpret rules, just ask them to break all your hard work as best they can. Trust me, if you have friends like mine, they almost certainly will dig up a few things you had not considered.
If you only want to play this with your friends as a house rule, this is where it ends. You take your class of Bob and you play and have fun.
If you want to release this to a wider audience then you can continue play testing with more people. Reach out to other groups in your area and ask then if they would play test for you and report back. Get out to your local cons and run scenarios and play tests with a more diverse groups of player.
This is not a thing I have developed in isolation and cannot claim credit for the mr_road method of class development, this is used by many games developers the world over.
Have an idea, and test it, test it, and test it again.
Also, listen to the feedback, it is hard, it is your baby and I know it is perfect in your eyes. But, it probably isn't and you need to listen to make it better. I always find this is the hardest part.