Does Prism calculate standard error of the geometric mean?

A standard error of the geometric mean would probably not be helpful. The standard error is expressed as a single number, which works for an ordinary mean because the uncertainty associated with estimating the mean is symmetrical about the mean itself. That's not true for the geometric mean, which is the antilog of the mean of the logs.

One suggestion: Convert all your values to base-10 logs. Compute the mean and 95% CI. Then take the antilog of the mean and both ends of the confidence interval. Those values are the geometric mean and its 95% CI. Note that the interval, while symmetrical on a log scale, is not symmetrical on a linear scale.