Which sentence is correct & why?
1. You can best handle the problem this way.
2. You can better handle the problem this way.
I think the first one is probably preferable. Remember the rule of when to use better and when to use best:
1. We use better when comparing only two things. In your sentences, you would use better if (and only if) there are only two ways to handle the problem.
2. We use best when comparing more than two things. In your sentence, if there are more than two ways to handle the problem, use best.
I think, however, that the wording of both sentences is a bit awkward. I'd simply say, "This is the best way to handle the problem."