The other answer is wrong, unfortunately.  Actually, that should be obvious, otherwise no statistician would ever bother to do SRS and they would happily stratify on utterly useless variables- why not?

It is true that the sum of squared error inside each stratum will drop (i.e. $\sum (x_i - {\overline{x}})^2$ will drop virtually always).  This is similar to the fact that adding new variables to a regression will just about always lower the R-squared even if the variable is complete noise.  However, the value of $s^2$ also involves dividing by $n_1 - 1$ and not by $n_i$.  In other words, if you have $k$ strata, you lose $k-1$ degrees of freedom.  So your estimated variances might rise if the stratification variable is sufficiently useless.  (The effect is most serious if sample sizes are small in some strata.  In the extreme, if they drop to $n_i = 1$, you can't even estimated the standard error.)

Crudely, a fast way to tell if stratification would help is to run an ANOVA on the stratification variable.  If it's significant (or, at least, the ADJUSTED R-squared rises), the stratification might help.

In practice, I tell researchers that as long as they have a reasonable sample size available from each strata, and the stratification variable makes sense (they are sure that means of one strata are 'significantly different' from those of other strata), then stratify.

Side note:  While lowering variances is the usual reason to stratify, there are others.  First, if you want to guarantee sufficient sample size in each stratum so that you can make separate inferences on each one, you should stratify.  Second, if costs vary greatly from one stratum to another, you can stratify to optimize costs.  Finally, if variability is known to be much higher in some strata than others, you can use stratification (by increasing sample size in the most variable groups) to lower your se.  However, if costs and variables don't distinguish your strata, you can definitely get a wider confidence interval if you stratify on an unhelpful variable.