On the one hand, it is true that those who seek out attention might stop committing crime to get that attention. Then again, maybe they would escalate their behavior in order to force the media to speak and to let word of mouth do the work for them. In a world of social media, trying to avoid bad news spreading by keeping it out of "the media" is an inherently ludicrous task. There are no gatekeepers anymore. But even in a pre-social media age, or if social media sites participated in this gagging of all bad news,
On the other hand, many of those who commit crime today would revel in the lack of attention they're getting. From serial killers to terror suspects, without the media reporting on them, the success rate at catching them would decline. The media help deal with crime
Both of these effects are noise-level compared to the bulk of crime. The fact is that the vast majority of crime is committed for much more pedestrian reasons than the desire to seek immortality in the media's memory. People rob, commit burglaries, vandalize, assault, commit battery, defraud, libel, rape and murder for reasons having almost nothing to do with media coverage of anything.
The drivers of crime, at least at a macro-economic level, are very well understood. There's lots of factors, ranging from environmental poisoning like lead (a huge variable in America in particularly) which impacts later behavioral traits to socio-economic variables like poverty and unemployment to the strength of mental health infrastructure, and while there's still a lot of subtlety to suss out and plenty of variables to further understand, we have a pretty robust set of variables that let us understand crime. The media just doesn't rank as a variable in this way. Direct studies of the media do show that more consumption of crime drama does tend to lead to a fear of crime, but the effect is usually weak; moreover, there is a cyclical effect, where someone already predisposed to be interested in and/or afraid of crime for whatever other reasons will watch crime dramas more readily.
Preventing media discussion of the underlying causes of crime, which would count as "bad news", and of crime rates would rob the public of the ability to have an informed response to crime. The media already does a spectacularly bad job there; reducing the amount of information to zero would only allow misinformation to spread even more rapidly and fear to be even more entrenched.
The problem with this question is that it assumes that people are basically drip-fed cows herded by the media. They're not. The media are certainly important and influential in the formation of culture and political attitudes, but people have an array of beliefs inherited from other sources, from their own experience to early peer group socialization to churches and social institutions. Those in turn impact the media too. The media is not some separate cloistered group of people giving commandments from on high: they're people who emerge from our society.
This year is in fact an object lesson on the relative powerlessness of the media to stop someone from using fearmongering and spreading misinformation about "bad news". Many media outlets are loudly indicating to people that crime is going down, and yet demagogues like Trump insist that we're in greater danger than ever. And while some of that can be blamed on conservative media, that too is a cyclical factor: people watch conservative media instead of the media telling the truth about these variables because they already had a conservative predisposition. Moreover, Trump has exceeded even FOX in rampant dishonesty and deliberate sociological illiteracy.
It's not that the media reports "bad news". It's that the media reports "bad news" in a specific way: without context, often with implicit or even explicit racial bias, in order to push ratings and often push specific agendas, leading to bad policies like the Three Strikes laws. If the media did that, we would be more able to deal with crime, yes, but it would still be incumbent upon the population to actually push for and craft sensible crime policy.