You can try connecting one of the analog inputs to the 5V supply and monitor/check it before reading the pots.
It all depends on how fast the voltage decays when power is lost (you’d like that to be relatively slow) and how fast you can read the pots and send-out the data after confirming the voltage is good (you’ll like that to be fast).
If you are running the Arduino off a higher voltage through the regulator, you can use a voltage divider (2 resistors) to bring that voltage down to 5V (or less) and monitor that voltage. That would give you more time because that voltage can drop before the 5V regulator drops-out, giving you more time to react.
Of course the problem is, when the voltage drops too far the processor will stop or get flaky and your serial transmission could get flaky. You need to know there’s a problem before that happens.
If you have any control on the receiving end, there may be things you can do to confirm the data is good. RS-232 has optional parity checking or you could do something more advanced. Ethernet has error checking, and I don’t know how the error checking/correction works but you rarely-rarely-rarely get corrupted data over the Internet even with data flowing all over the world through all kinds of unknown/flaky connections.
Maybe you can send the data twice and make sure it’s the same twice? Not efficient, but it might work…