Welch's t-test when the smaller sample has a larger variance I have heard that there is no statistical criterion, which would work well in the case when the smaller sample has a larger variance due to the Behrens-Fisher problem, and even Welch's t-test can't be used in this situation. 
Is it true? I can't see such limitation in Wikipedia. 
 A: As far as type I error rate goes, it looks okay to me.
A few quick simulations don't seem to indicate any major difficulty with significance level when the smaller sample has the larger variance.
For example, with the smaller sample having $n_s=10$ and the larger sample with $n_l=100$, and with the smaller population having 25 times the variance of the larger (5 times the sd), the actual significance level when the test is carried out as a 5% test is very close to 0.05 (this is R code):
> sr=5;res=replicate(10000,t.test(rnorm(100),rnorm(10,0,sr))$p.value)
> mean(res<.05)
[1] 0.0501
> sr=5;res=replicate(10000,t.test(rnorm(100),rnorm(10,0,sr))$p.value)
> mean(res<.05)
[1] 0.0499

That seems pretty good. Of course you'd want to try a range of n's and ratios of variances. I tried a few others, but they were all reasonably close to 5% for a 5% test - e.g. a couple of trials of 10000 simulations for a ratio of variances of 9 and a ratio of sample sizes of 100/10 gave the type I error rate at about 5.1%. Four such trials with the ratio of variances changed to 100 gave an average type I error rate of 4.9% -- but the standard error of the estimate is about the same size as the deviation from 5% so we can't actually pick up any deviations from 5% here (there will be some, they're just too small to pick up even with 40000 trials). A much larger trial suggests that it's actually closer to 5% than this.
I'd be quite untroubled by such small deviations in the type I error rate; if you're concerned about some particular $n_s, n_l$ and some range of variance ratios, you could easily simulate to check that case in detail, but I'd suggest that apart from some possible issues at very small sample sizes ($n_s=2$ might might have a few poor ones, say, so you'd probably want to check that more carefully if you had $n$ so low), that pretty much all cases you're likely to encounter in practice will generally be okay.

What about power? It depends on what you compare it to. If the problem is only pairing larger variance with smaller sample, let's look at the effect on asymptotic precision (inverse variance) of swapping the larger variance to the larger sample; this should tell us approximately how much larger a pair of samples we'd need to pick up the same small difference in means.
It certainly will have an effect, since in larger samples, the power is most heavily determined by the smaller of the two precisions of the group means; the case where the bigger variance goes with the smaller sample size will certainly reduce power because of that. 
Let the ratio of sample sizes be $n_r=n_l/n_s$ and as before let $V_r$ be the ratio of variances (larger on smaller). Also, for this calculation, let the smaller variance be 1.
In the case you're asking about the precision of the difference in means is proportional to $n_s(1/V_r+n_r)$ and in the case where the larger variance is with the larger sample it's proportional to $n_s(1+n_r/V_r)$
Further, let's look at the simple case where $V_r=n_r$ e.g. consider 9 times the sample size, and 9 times the variance. Then the ratio of the two precisions is 4.55 --- that is, the effective precision will be about 22% of what it would be with the larger variance combined with the larger sample size. Which means to get about the same power at the same ratio of sample sizes you'd need 4.5 times as many data points when the variance is with the smaller group. (Of course if you can control it you can improve things much more quickly by focusing on improving the smaller sample size.)
[Why did I set those ratio to be equal? As they become more different it's the larger of the two ratios that mostly drives the precision of the difference, and hence the asymptotic relative efficiency; the case where they're close to equal is the interesting region, that's where you get the relative impact of swapping the variance between the smaller and the larger sample size to be big]
This doesn't make the test unusable, just relatively less powerful. It's not like there's all that much to be done about that, since there's actually less information about the difference.
