It's recommended to turn off brown-out detection (of an MCU) if it consumes too much energy for your application.
For example, with the ATmega328P, the MCU typically only consumes 0.1 µA during power-down sleep (where the contents of the RAM/registers is still conserved), but its power consumption increases to 17 µA when BOD is enabled.
Depending on the type of battery this might be higher than self-discharge effects and thus worthwhile to avoid. I'm not an expert on battery chemistry, but e.g. 10 µA self-discharge equivalent seems to be plausible with alkaline batteries.
When the MCU's BOD is switched off you can still detect brown-out externally.
For example, you can use an LDO voltage regulator with enable pin feature such as the RT9080 and combine it with a push/pull voltage detector (PMIC supervisor) such as the MCP112 or TC54.
In comparison with the ATmega328P internal BOD, the above voltage detectors are more efficient (e.g. I_DD_typ=1µA and I_DD_max<4µA).
Alternatively, with the ATmega328P there is also a middle ground. That means it's possible to just disable the BOD during power-down sleep mode.
This trades off wake-up latency for lower power usage during sleep. (i.e. 60 µs or so additional wake-up overhead).
Using it like this still has reasonable safety properties because the MCU automatically re-enables BOD during wake up.
Also, you might prefer to use an external BOD that simply switches off the MCU, because you want to protect your battery from deep discharge, anyway.
For example, the BOD of the ATmega328P puts the MCU into reset mode when it detects a brown out condition. And there it consumes 1 mA or so such that your battery continues being discharged at that rate.
Also, all configured pull-up resistors are disabled during reset, such that you have to connect all unused pins to external pull-up/pull-down resistors to avoid additional current consumption because of floating pins.