Are human beings responsible for making good use of their talents, or is stagnation acceptable?

I assume you are asking what position Hamlet takes on this question. As with any play, it is more difficult to tell the perspective of the author on any topic than with a novel, because there is no narrator to provide an angle on the events. However, the character Hamlet himself certainly comes to believe that humans have a responsibility to use their talents rather than simply continuing to live. In his act 4 soliloquy, generally referred to by its first line—"How all occasions do inform against me"—Hamlet says:

What is a manIf his chief good and market of his timeBe but to sleep and feed? A beast, no more.Sure He that made us with such large discourse,Looking before and after, gave us notThat capability and godlike reasonTo fust in us unused. (4.4.35-41)

In other words, it's not good enough to continue living, to see mere eating and sleeping as the whole point of life. God provided mankind with reason, says Hamlet, and humans must use that "godlike reason" to decide what is important and do that. To fust means "to become moldy through disuse" (Merriam-Webster). So Hamlet is saying that we can't just sit around and let our reason get moldy—metaphorically, of course—because we never use it. God didn't give us "that capability" to think so that we could just waste it on sitting around and doing nothing. By act 4, at least, Hamlet personally believes that we have an obligation to act. While we don't know for sure what Shakespeare thought, his putting these words in the mouth of his main character suggests that he at least considered this perspective important.
https://www.merriam-webster.com/dictionary/fust