Someone on Nextdoor was asking this question, and I was tempted to respond that the only thing HOAs collect is money. The real responses to the question ranged from, "It depends on your HOA" to "That's not part of what an HOA does." While I can understand that crime is separate from house paint and green lawns, since crime is the #1 factor in property values, HOAs probably need to DO something when it comes to crime.
I read that in a town in the Southwest, an HOA was being sued because a house that had been deemed a nuisance was not foreclosed on, resulting in a child being killed in a drive-by shooting meant for the residents of the nuisance house. I agree with the people bringing the lawsuit. If the HOA's sole purpose is to maintain property values, then people who bring crime to the neighborhood should be forced out, just like someone who might never mow the grass or pay HOA dues.
When one of our neighbors brought two violent incidents to our block, I wrote to my HOA and to the owner of the rented house. I mentioned the lawsuit above to the HOA, and I told both entities that other neighbors had confirmed to me that the police showed up at the house several times before we'd moved in. I myself had seen a member of the household arrested a couple months before the violence. So three incidents in about four months, and I decided it was time for the family to move on. I never really received much of a response, but the family DID move out within a couple of weeks. Maybe the family realized it was time for a fresh start, but I'd like to think that getting involved helped to push them. However, I also believe the HOA and rental company would have either not known about it or ignored the problem without my letters.
Crime, however, is often a result of people from outside of the neighborhood. That crime itself may not be the responsibility of the HOA, but installing a trail camera on a cell phone plan with a solar power battery pack
might do the trick to capture trespassers. Any kind of camera, really. See if the police will install one. Even a fake camera might deter one crime. I'd let a wifi camera on my street use my guest wifi, as well, and I assume some of my other neighbors would, too.
The HOA could go semi-all-out and install an automatic gate without a code, along with a powered camera that stores video. If there's no code, anyone can get into the neighborhood, but it slows people down and ensures everyone entering knows they are on camera. The good news is that the roads are still public and part of the city.