Why do I measure voltage on a breaker that's off?

Almost certainly an induced voltage. Is this part of a Multi Wire Branch Circuit (MWBC)? If you have black, red & white wires in the cable it's probably a MWBC. Common to have induced voltages in them.

In a lot of homes in the US, the circuits for your dishwasher and garbage disposal are fed to a single duplex receptacle under the sink, which has the two hot wires coming from separate breakers to the separated receptacles, but they share the same neutral wire (called a Multi Wire Branch Circuit, or MWBC). What you are reading, because you most likely have an inexpensive digital meter, is just the capacitive coupling though the motor windings and starting capacitors of those motors because they are sharing the neutral. If you had used an older meter with an inductive coil on it or a more expensive digital one that has proper filtering, the meter burden would have taken that voltage reading to zero. Bottom line, don't worry about it.

You have a multi-wire branch circuit feeding these two appliances. This means that it's a single /3 cable, with two hots and a shared neutral. Since the two hots run together in the same cable, you are getting capacitive "crosstalk" between the two wires. This is very weak, and you can only see it because you're using a DVM. DVMs are very sensitive.

The bigger problem is the breakers

And the biggest risk is in phasing the multi-wire branch circuit incorrectly, which will overload the neutral wire. The surest way to get that right is to use a factory provided handle-tie, either built in to a 2-pole breaker, or made by the factory to tie two individual breakers. The handle-ties are keyed so they won't fit on the breaker in a mis-phase situation. 2-pole breakers are much easier to obtain. Note that you will need a 2-pole breaker if you ever want to install GFCI protection.

Second, it should not be possible to turn only half the circuit off. That is precisely to protect you from getting nailed like you almost did. The 37 volts wouldn't have harmed you, but if you interrupted the neutral, that would've! Bigtime. Interrupting a neutral on a live circuit can kill you, that's why neutral wires have insulation.

So, when you use a 2-pole breaker or factory handle-ties to tie the two breakers together, that takes care of that also.

Again, the reason not to tie them with a nail is because a nail will let you tie, say, the two halves of a duplex breaker, which is something you should not do. That would cause precisely the overloaded-neutral scenario I mentioned.