The simple answer to your question is that the characters in Measure for Measure are “problematic” because they are found within one of Shakespeare’s few “problem plays.”
The term “problem play” was coined by a literary scholar named Boas who ascertained that there were a few plays in the middle of Shakespeare’s writing career that bridged the gap between his original comedies and the darker work of his tragedies. As a result, these few plays shift dramatically in tone throughout, causing issues for the characters (and the actors). The list of these “problem plays” are as follows (says Boas): Measure for Measure, All’s Well That Ends Well, Troilus and Cressida, and Hamlet as the true bridge. Further, Boas says the following:
Hamlet, with its tragic close, is the connecting-link between the problem-plays and the tragedies in the stricter sense.
Now, why are they considered “problem plays”? Because even though they end on a general note of happiness, they have deeper and darker issues that can never be resolved.
Shakespeare’s Measure for Measure certainly does end on a positive note for many characters, thus the characters can be considered “problematic.” The play ends with many marriages and engagements. The Duke Vincentio ask Isabella to marry him. Claudio does, indeed, marry the pregnant Juliet. Mariana, originally forced to marry Angelo and in an extraordinary display of compassion, begs Angelo’s pardon from the Duke. Further, the Duke Vincentio (disguised as the Friar, mind you) convinces the people that the government represented by Angelo is bad and that the original ruler (the Duke) is the preferred ruler. This sounds like a comedy, doesn’t it? Many people get married, and the country is safe and sound.
However, Shakespeare’s Measure for Measure has many dark and deep issues that are unable to be resolved by the various characters which could also be considered “problematic.” First of all, how about the idea that Duke Vincentio has to resort to a scheme to reenact enforcing the rules? Yes, he is a temperate ruler, but as a result, a “deputy governor” (Angelo) needs to do the dirty work in order to bring the community back from disrepair. Next, we have the case of identity issues. How appropriate is it for Duke Vincentio to disguise himself to see that the job gets done? Further, Duke Vincentio decides not to simply disguise himself as a citizen, but as a member of the clergy. It seems to me that this shows some real trust issues between Duke Vincentio and even his most trusted advisors.
Our doubts are traitors, / and make us lose the good we oft might win, / by fearing to attempt.
Or how about this for a deep and dark issue: is it truly the best thing for Angelo to suggest killing a young man for impregnating his fiancé? Even in political terms, wouldn’t the child become the ward of the state? Yet another deep and dark issue is Angelo’s lustful desires towards Isabella (Juliet’s sister). He asks her to lose her virginity to him in exchange for the young man’s life. Or how about the cowardly brother, Claudio, who simply suggests that Isabella placate Angelo to save a life? Do you see how these are truly deep and dark issues that cannot be resolved? All of these convoluted ideas and activities escalate further until it seems that there can be no solution. The irony is that there is, in fact, a solution; however, it is a solution that doesn’t truly solve any of the underlying issues.
The tempter or the tempted, who sins most?
In conclusion, the characters in Shakespeare’s Measure for Measure can be seen as “problematic” because the play itself is considered a “problem play” by such scholars as Boas. I suppose it is possible, however, that you mean the characters are problematic in other ways as well such whether they are good people with real issues, or evil people with positive characteristics. In this regard, I think the most “problematic” character would be Duke Vincentio. Although he is a compassionate and even temperate ruler, Duke Vincentio knows that he is at least partially to blame for all the troubles in his community because he has failed to enforce the law. As a result, he resorts to scheming in order to perk up the discipline. Positive intentions, questionable methods.