The Travel Health Pro website (funded by the British Government to provide accurate information to travellers and health professionals) confirms that there's no risk of Yellow Fever in the former Netherlands Antilles:
https://travelhealthpro.org.uk/country/44/netherlands-caribbean-islands#Vaccine_recommendations
Yellow Fever vaccine is produced from an
attenuated virus. It can't harm your boyfriend (unless he's got a severely immunosuppressed system, as with some HIV patients) and it can't harm anyone who is bitten by a mosquito which has already bitten him.
Indeed, if your boyfriend had been travelling Bonaire from a country where Yellow Fever exists, the authorities would INSIST that he'd had the vaccination before allowing him to enter. (They'd hardly have such a rule if they regarded having a vaccinated person in their country as a threat to others around him!)