Your error bars may be giving you a hint to look more closely at the distribution of your data. For example, if your data is essentially log-normal you could work with the logs of your numbers and the problem will automatically go away.
I'm not a fan of error bars. They ideally give a sense of underlying distributions and 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 are arguably as a better way to represent asymmetric distributions, along with text/captions that highlight the important statistical significance conclusions.
