Well, assuming Mind Control allows the villain to essentially say "now enter your PIN" without knowing the PIN, that would make any sort of electronic banking, such as an ATM or Internet banking absolutely a no-go. So we're now reduced to nothing but physical locations only. If not, then I guess the easiest solution is to just give everyone their own super secret passwords that they memorize, and now the bad guys would have to figure out the victim's password the same way that they do in our world.
Can two people attempt to mind-control someone at the same time? IE, could the bank have their own mind-controller who can take over a customer's mind strictly to ask "Are you making this decision of your own volition?" because that would be my immediate counter to a bad-faith mind controller.
Teleportation and invisibility are both nice, but they're also both countered fairly easily-- just put weight sensors on the floor in the places where you don't want them. If the weight sensor is triggered when there isn't supposed to be anyone in there, then it sets off an alarm, seals the area, and maybe sprays something like fluorescent/smelly paint at the area where the sensor just went off. Now it's going to be difficult/impossible for the culprit to escape justice (we have semi-similar measures for riot police to track protestors in the real world today; Chinese MPs have recently used them against protestors in Hong Kong).
To counter teleporters, maybe the vault area is irregularly shaped? IE, it's sort of "ribbed," with random sections of the wall jutting out, random areas curving back in, etc. From the outside, it's a box. On the inside, it's a series of small chambers that maybe don't use space super efficiently, but make it incredibly dangerous to try to teleport in. Or perhaps money and such are all stored in rooms that are almost exactly the size of whatever needs to be stored in them? If there's less than one human's worth of space behind a locked door, a teleporter will have a very hard time jumping in there without teleporting their leg or body into a physical object and losing it.
Another counter-measure to make sure that an invisible person doesn't just tag along and stick very close to a person who should legitimately be in the vault-- spray the area down with a cheap gas, like steam. Being invisible won't stop the solid from bouncing off of their person, and it would reveal any sneaky individual trying to make their way into the vault.
Super strength has probably the absolute easiest solution-- do what we do to take care of regular strength. Hire other strong guys. If one strong man wants to try to take on a bank with regular security guards, he can just do that. If that bank has eight security guards with super strength wandering around, he's gonna need a super-powered criminal posse. Maybe the bank even hires a mind-control person or two; just counter the bad guys on your own terms. And the nice thing about them is that they don't need to be (and likely shouldn't be) obvious to the public. They can just look like normal schmucks going about their duty, or they can even serve double-duty as the tellers and money-movers (a super strong guy would be great to have around when you need to transport physically heavy items from one point to another!)
This all obviously costs money for the bank, so it's unlikely that you'll see as many banks as we do now-a-days. You'd probably see only a few per city, with only a few branches for every few thousand people. Rural communities would be hit the hardest by this, as anyone wishing to drop off a paycheck or pick up cash for the week will have to travel into the nearest town to get that advance. Hope you saved enough cash to get gas on the way into Indianapolis, Farmer Ted. It's a three hour drive each way from PoDunk, and now you have to do it once a month! Maybe we'd see the advent of a barter system from these people, as they wouldn't want to be bothered so darned much? But that's besides the point....
The final issue here is determining that you get good, honest supers to counter the bad supers. The bank's biggest risk is the people who are already on the inside; if those supers turn on them, there's not a whole lot that can be done to stop them. Easiest way around that is to just hang onto their personal data, same as we do in the real world. SSN, photo ID, copy of the birth certificate, etc.