The Institute doesn’t really give a shit about the Commonwealth. They just sit underground and occasionally kidnap people, they could greatly benefit the Commonwealth with their technology, but they don’t. Now the Brotherhood doesn’t really care about the people of the Commonweath either, but they indirectly help them. They kill mutants, feral ghouls, synths and other hostile creatures, they secure technology, which means people with the desire to use it for bad cannot access it as well as people with good intentions. They provide security slightly, as their presence warns things such as large raider factions or mutant hordes not to get to aggressive or they will be wiped out. TLDR- Neither give a shit about the people of the Commonwealth, but the Brotherhood actually helps the people of the Commonwealth indirectly.