I have some statistical data from which I want to graph the means and use the standard deviations as error bars. However this produces a graph with some of the error bars passing below zero. A negative value is silly for this data (mean trip times), so I was wondering what is a sensible way to graph the data.

Your error bars may be giving you a hint to look more closely at the distribution of your data: it may not be symmetric. For example, if your data is essentially lognormal you could work with the logs of your numbers and the problem will automatically go away. I'm not a fan of error bars. In theory they let you visually do some statistical significance estimates and perhaps give some sense of the underlying data. But there are a lot of subtleties and at least one study has found that even experienced scientists often misinterpret them. This nice blog post discusses some of the issues. If you do need to summarize the data with a few statistics, I'd argue for boxplots as a better way to represent asymmetric distributions, along with text/captions that highlight important statistical significance conclusions. 


Perhaps means and standard deviations are the wrong way to present the data. It sounds like you would communicate more information if you graph the medians and used quartiles as "error bars". 


Assuming you mean standard deviation, if the error bars are $\mu \pm \sigma$, and $\mu  \sigma < k$, where $\mu$ and $\sigma$ are mean and standard deviation respectively, and $k$ is the minimum value taken by your random variable, you can always leave the error bars at $k$ and $\mu + \sigma$. Similarly for $\mu + \sigma > K$, where $K$ is the maximum value your random variable takes. 


Probably its very late answer for the actual person. But I am going to suggest an answer anyways so that others can use this formation. "You can calculate standard error (SE) for the data and include them as error bars and they should not go below zero". SE = Standard deviation / SQRT(sample size). SE represents for the population rather than sample so it will be more informative than Standard deviation. 

