Are infamous assassins bad and are they more likely to be caught? Is it better to have an assassin who rather unknown than one who is infamous.
This applies for almost any professional criminal: Their job will be easier and safer if no one knows who they are.
Realistically, being a world famous assassin or thief would be a nightmare. They’d be the first suspect whenever anything high profile happened in their general vicinity. They’d also be under tighter scrutiny, if they went anywhere publicly, they’d be carefully watched. (Either by the authorities, security, or both.) It would be significantly harder, or impossible, for them to do their job.
There’d be a real risk of being implicated in crimes they had nothing to do with. Fabricating a case against a famous assassin or thief would even be an effective tactic for anyone who wanted to neutralize them (either for revenge or to further their ambitions.)
It’s also, legitimately questionable how these characters could build up infamy. Murdering someone is, as you may be aware, somewhat frowned upon by polite society. So, if you have a character who’s built their reputation by murdering people, it raises the question, “How have they avoided the consequences?” The real answer would be, “by avoiding detection,” but being sneaky and escaping attention is the opposite of how one builds a reputation.
In the real world, there’s debate whether any, “master-class,” assassins actually exist. The entire nature of the job requires that they avoid detection, and if they’ve done so successfully, they’re effectively unknown. The assassins we do know about are the amateurs, and low-skill professionals, because they are identified and prosecuted. In that sense, becoming infamous as an assassin is somewhat analogous to being bad at your job.
Things could be a little different if your setting has some kind of legalized assassins guild. “World famous,” assassins are a staple in certain flavors of fiction. Some kind of secret society of assassins would give its members a safe space to develop their reputations, but they would be virtually unknown outside of their organization, (and groups it associates closely with.)
So yes, being famous (or infamous) would make an assassin’s job nearly impossible.