Lower variance in the predictor leads to larger standard errors - when the predictors are orthogonal, they are exactly inversely proportional in a least squares model, as can be seen from the well known formula:
$$
{\rm var}(\hat\beta_{j}) = \sigma^2[(X'X)^{-1}]_{j}
$$
where $\sigma^2$ is the error variance and $X$ is the design matrix. Similarly, the standard errors in a GLM are generally inversely related in GLMs like a logistic model. In the extreme case where you have no variance in the predictor, the effect is not estimable and you will get an error when you attempt to fit the model.
As an example, consider logistic regression with a single predictor $X_{i} \sim N(0,\sigma^{2})$:
$$ \log \left( \frac{ P(Y_{i} = 1) }{ P(Y_{i} = 0 } \right) = \beta_{0} + \beta_{1} X_{i} $$
In the code below I simulate from the model under increasing values for $\sigma^2$ and show that the standard error decreases. In all simulations $\beta_{0} = 0$, $\beta_{1} = 1$, $n = 1000$. $\sigma^{2}$ is incremented from .1 to 2 in such a way that there are 1000 points. The empirically observed standard errors from a single set of simulations are plotted below. The apparent "bumpyness" in the plot in monte carlo error - bump up the sample size and that will go away.
s = seq(.1, 2, length=1000)
V = rep(0,1000)
for(i in 1:1000)
{
x = rnorm(1000,mean=0,sd=s[i])
y = (x + rlogis(1000))>0
g = glm(y ~ x, family="binomial")
V[i] = summary(g)$coef[2,2]
}
plot(s,V,pch=16,xlab="Variance of the predictor",ylab="Standard error of regression coefficient", cex.lab=1.5, cex.axis=1.5)
