Public opinion is important in a democracy because we are the "employers" of those whom we elect. They serve at our behest, so what we think about what they do should matter a great deal to them. Every few years, we issue a performance review, by way of election or not. However, we seem to be in a period in which our elected representatives are slaves to polling. Whether this is good or bad is a function of one's ideas and expectations about representational governance and decision-making.
Some people think that when we elect representatives, they should act in accordance with our wishes on all matters. So that if a majority in a congressional district opposes a trade agreement, there is an expectation that the representative will vote against it. If a majority in a district seek to defund Planned Parenthood, the representative should vote accordingly.
Some people think that we elect representatives to make decisions using their ethics, knowledge, intelligence, and experiences to make the best choices possible for those whom they represent. Thus, we speak of a senator casting a vote or taking a stance based upon his or her own conscience. Or a representative will vote for a bill because he or she has investigated it thoroughly and knows far more than the people in the district know about it.
For both approaches, public opinion matters, of course. If public opinion turns too greatly on a politician, he or she is voted out of office. In the first case, public opinion informs the decision to be made, based upon what a majority want the elected representative to do. In the second case, the representative needs to know how deeply unpopular a decision might be, to weigh the costs of an unpopular decision.
The problem today is that in the 24/7 world of news that we live in, there should be legitimate concern about what people are basing their polling responses upon. The latest soundbite, true, false, or as some say, "truthy," can probably sway millions of people to a completely different opinion. If this continues, polling is likely to become increasingly inaccurate. No pollster can keep up with so many shifts. Politicians themselves are responsible for this problem, too, since they frequently change their positions. I personally think that people running for office before the advent of the internet were more likely to state what their positions were consistently, not being able to send up trial balloons so easily and then walk them back after a negative public response. Technology also gives us all these new means of viewing public opinion, tweets, Facebook posts, and so on, ad nauseum. Perhaps a Facebook post should be accorded more weight than a tweet, so that five negative tweets equal one Facebook post. How ridiculous can all of this get as politicians seek to take the public's temperature?
No matter what you believe about how a representative should be representing you, it is clear that what the public thinks matters to our elected representatives. They need to know how their constituencies want them to vote, and if they vote contrary to their constituency's wishes, they need to be able to calculate the political cost.