Without wishing to sound in any way evasive, I sympathize with both characters. I sympathize with Bartelby, because it's clear that there's something wrong with him; I also sympathize with his employer because he has a business to run and doesn't know how to handle this strange individual holding up the work of the entire office. However, if push came to shove, I'd say that my main sympathies are with Bartleby, as he appears to be experiencing some kind of mental health issue. Whatever the precise diagnosis, it's clear that Bartleby's pretty much given up on life, and for whatever reason no longer feels that he can go on.

With regards to how he should've been handled, that's a difficult question. When Melville wrote the story, mental illness was poorly understood. Individuals with mental health issues tended to be neglected; they were kept out of sight and out of mind in squalid institutions. Sometimes, such poor unfortunates would end up being slung in jail, and this is what happens to Bartleby. Nowadays, the chances are that Bartleby would be able to get the help and support he so obviously needs, but in those days it was a different matter entirely.

Sympathy with characters is one way of reacting to a work of fiction, but it is not either the only way of reacting nor are all stories designed to elicit such reactions. In this case, the character of Bartleby, as presented through the lens of the affable but not profound narrator, appears more of a mysterious and enigmatic character that serves as a litmus test of the reactions of others than a fully-rounded character. While I find the relationship of the narrator to his employees interesting, and appreciate Melville's artistry in making the narrator's fascination with Bartleby credible, the story to me seems designed to evoke reflection rather than empathy.

The narrator appears a basically morally good character and good boss. He allows for the foibles of others, and goes out of the way to treat his employees well. His style of management is paternalistic, and while he acts out of self-interest, he is also generally benevolent.

Personally, as an employer, I probably would have replaced Bartleby much earlier, rather than tolerating him for so long and moving offices to avoid him.