"Jesse Custer." He introduced himself to both of them. "I haven't lost no one, not yet. I guess ya could say my mournin' is due to the fact that I can't do nothin' to help."
If he were a less moral-filled guy, maybe he'd ignore it all. Or maybe if he knew that there wasn't anybody in this place that The City could go after to punish him, he'd do it. Getting himself hurt or in trouble was one thing, getting somebody else in the same situation was entirely different.
All he could think of was poor Mel having to deal with some huge trauma because he decided to get rid of the killer.
"I don't know your Karen," Jesse told Venkman "But I think you got as much right as any to mourn her loss. Just cause you ain't known somebody very long don't mean you don't care for 'em."