Not important at all. The star system at Morningstar is predicated on how the fund has done historically. You’re probably not interested in betting on a horse that ran the Kentucky Derby in 2011, right? Morningstar is working on improving its ratings system and has begun incorporating more forward-looking criteria, but it’s a work in progress. As for fees, they are one of the investor’s natural predators along with taxes. There are very few funds in this world, 5 star or otherwise, that are worth paying up for.
I frankly don't pay any attention to the Morningstar stars, though as Julian mentioned they have changed their fund ranking system so it will be interesting to see how this system works as a predictor of future perfomance.
Low fees are a great starting point. As one who starts with an asset allocation plan (as do many of my advisor peers) I look for opportunities to implement that allocation with low cost index funds/ETFs where it makes sense. With an index fund cost and adherence to the underlying index are all that counts.
Beyond that when and where I use actively managed funds, expenses are still important. Not so much that the fund needs to be the cheapest in its peer group, but I find that most of the active funds that I use have competitive expenses. I've not found a fund manager who is good enough to consistently justify costs that are well above average.
Star ratings offer a good view of what has already happened. It is useful to evaluate a fund's past performance and understand why it did or did not do well, but it may not be as useful in trying to determine how a fund will perform going forward. Low fee funds, like index funds, may be appropriate for investors who don't want to do a lot of analysis, and a number of these will have a higher star ranking. But buying an index fund or two is not much different than buying a 5-star fund because the purchase decision misses the point of all the questions you should ask BEFORE you consider what funds to buy. You should develop an investment plan based on your specific goals and time frame, and only then evaluate which asset classes and combinations of asset classes will best serve you. The choice of funds is subservient to these decisions. Don't put the cart before the horse; do your planning first. PS - low fees are good, but they are not the only consideration.
Star rating change...Morningstar even just changed their rating system to reflect the underlying process of funds rather than just rank on a star system. I feel underlying fees are more important when comparing index funds...they essentially are holding the same investments, i.e. the S&P 500, so in that example you would be best served by choosing the fund or ETF (if you have access to ETF's) with the lowest fee as you are paying for passive management, not active management.